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Theorem List for Metamath Proof Explorer - 32901-33000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorembnj528 32901 Technical lemma for bnj852 32933. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  G  =  ( f  u.  { <. m ,  U_ y  e.  ( f `  p )  pred ( y ,  A ,  R )
 >. } )   =>    |-  G  e.  _V
 
Theorembnj535 32902* Technical lemma for bnj852 32933. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph'  <->  ( f `  (/) )  = 
 pred ( x ,  A ,  R )
 )   &    |-  ( ps'  <->  A. i  e.  om  ( suc  i  e.  m  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  G  =  ( f  u.  { <. m ,  U_ y  e.  ( f `  p )  pred ( y ,  A ,  R )
 >. } )   &    |-  ( ta  <->  ( ph'  /\  ps'  /\  m  e.  om  /\  p  e.  m ) )   =>    |-  ( ( R 
 FrSe  A  /\  ta  /\  n  =  ( m  u.  { m } )  /\  f  Fn  m )  ->  G  Fn  n )
 
Theorembnj539 32903* Technical lemma for bnj852 32933. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ps 
 <-> 
 A. i  e.  om  ( suc  i  e.  n  ->  ( F `  suc  i )  =  U_ y  e.  ( F `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  ( ps'  <->  [. M  /  n ].
 ps )   &    |-  M  e.  _V   =>    |-  ( ps'  <->  A. i  e.  om  ( suc  i  e.  M  ->  ( F `  suc  i
 )  =  U_ y  e.  ( F `  i
 )  pred ( y ,  A ,  R ) ) )
 
Theorembnj540 32904* Technical lemma for bnj852 32933. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ps 
 <-> 
 A. i  e.  om  ( suc  i  e.  N  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  ( ps"  <->  [. G  /  f ]. ps )   &    |-  G  e.  _V   =>    |-  ( ps"  <->  A. i  e.  om  ( suc  i  e.  N  ->  ( G `  suc  i
 )  =  U_ y  e.  ( G `  i
 )  pred ( y ,  A ,  R ) ) )
 
Theorembnj543 32905* Technical lemma for bnj852 32933. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph'  <->  ( f `  (/) )  = 
 pred ( x ,  A ,  R )
 )   &    |-  ( ps'  <->  A. i  e.  om  ( suc  i  e.  m  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  G  =  ( f  u.  { <. m ,  U_ y  e.  ( f `  p )  pred ( y ,  A ,  R )
 >. } )   &    |-  ( ta  <->  ( f  Fn  m  /\  ph'  /\  ps' ) )   &    |-  ( si  <->  ( m  e. 
 om  /\  n  =  suc  m  /\  p  e.  m ) )   =>    |-  ( ( R 
 FrSe  A  /\  ta  /\  si )  ->  G  Fn  n )
 
Theorembnj544 32906* Technical lemma for bnj852 32933. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph'  <->  ( f `  (/) )  = 
 pred ( x ,  A ,  R )
 )   &    |-  ( ps'  <->  A. i  e.  om  ( suc  i  e.  m  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  D  =  ( om  \  { (/) } )   &    |-  G  =  ( f  u.  { <. m ,  U_ y  e.  ( f `  p )  pred ( y ,  A ,  R )
 >. } )   &    |-  ( ta  <->  ( f  Fn  m  /\  ph'  /\  ps' ) )   &    |-  ( si  <->  ( m  e.  D  /\  n  = 
 suc  m  /\  p  e.  m ) )   =>    |-  ( ( R 
 FrSe  A  /\  ta  /\  si )  ->  G  Fn  n )
 
Theorembnj545 32907 Technical lemma for bnj852 32933. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph'  <->  ( f `  (/) )  = 
 pred ( x ,  A ,  R )
 )   &    |-  D  =  ( om  \  { (/) } )   &    |-  G  =  ( f  u.  { <. m ,  U_ y  e.  ( f `  p )  pred ( y ,  A ,  R )
 >. } )   &    |-  ( ta  <->  ( f  Fn  m  /\  ph'  /\  ps' ) )   &    |-  ( si  <->  ( m  e.  D  /\  n  = 
 suc  m  /\  p  e.  m ) )   &    |-  (
 ( R  FrSe  A  /\  ta 
 /\  si )  ->  G  Fn  n )   &    |-  ( ph"  <->  ( G `  (/) )  =  pred ( x ,  A ,  R ) )   =>    |-  ( ( R 
 FrSe  A  /\  ta  /\  si )  ->  ph" )
 
Theorembnj546 32908* Technical lemma for bnj852 32933. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  D  =  ( om  \  { (/)
 } )   &    |-  ( ta  <->  ( f  Fn  m  /\  ph'  /\  ps' ) )   &    |-  ( si  <->  ( m  e.  D  /\  n  = 
 suc  m  /\  p  e.  m ) )   &    |-  ( ph'  <->  ( f `  (/) )  = 
 pred ( x ,  A ,  R )
 )   &    |-  ( ps'  <->  A. i  e.  om  ( suc  i  e.  m  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   =>    |-  ( ( R  FrSe  A 
 /\  ta  /\  si )  -> 
 U_ y  e.  (
 f `  p )  pred ( y ,  A ,  R )  e.  _V )
 
Theorembnj548 32909* Technical lemma for bnj852 32933. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ta 
 <->  ( f  Fn  m  /\  ph'  /\  ps' ) )   &    |-  B  =  U_ y  e.  (
 f `  i )  pred ( y ,  A ,  R )   &    |-  K  =  U_ y  e.  ( G `  i )  pred (
 y ,  A ,  R )   &    |-  G  =  ( f  u.  { <. m ,  C >. } )   &    |-  (
 ( R  FrSe  A  /\  ta 
 /\  si )  ->  G  Fn  n )   =>    |-  ( ( ( R 
 FrSe  A  /\  ta  /\  si )  /\  i  e.  m )  ->  B  =  K )
 
Theorembnj553 32910* Technical lemma for bnj852 32933. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph'  <->  ( f `  (/) )  = 
 pred ( x ,  A ,  R )
 )   &    |-  ( ps'  <->  A. i  e.  om  ( suc  i  e.  m  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  D  =  ( om  \  { (/) } )   &    |-  G  =  ( f  u.  { <. m ,  U_ y  e.  ( f `  p )  pred ( y ,  A ,  R )
 >. } )   &    |-  ( ta  <->  ( f  Fn  m  /\  ph'  /\  ps' ) )   &    |-  ( si  <->  ( m  e.  D  /\  n  = 
 suc  m  /\  p  e.  m ) )   &    |-  C  =  U_ y  e.  (
 f `  p )  pred ( y ,  A ,  R )   &    |-  G  =  ( f  u.  { <. m ,  C >. } )   &    |-  B  =  U_ y  e.  (
 f `  i )  pred ( y ,  A ,  R )   &    |-  K  =  U_ y  e.  ( G `  i )  pred (
 y ,  A ,  R )   &    |-  L  =  U_ y  e.  ( G `  p )  pred (
 y ,  A ,  R )   &    |-  ( ( R 
 FrSe  A  /\  ta  /\  si )  ->  G  Fn  n )   =>    |-  ( ( ( R 
 FrSe  A  /\  ta  /\  si )  /\  i  e.  m  /\  p  =  i )  ->  ( G `  m )  =  L )
 
Theorembnj554 32911* Technical lemma for bnj852 32933. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( et 
 <->  ( m  e.  D  /\  n  =  suc  m 
 /\  p  e.  om  /\  m  =  suc  p ) )   &    |-  ( ze  <->  ( i  e. 
 om  /\  suc  i  e.  n  /\  m  = 
 suc  i ) )   &    |-  K  =  U_ y  e.  ( G `  i
 )  pred ( y ,  A ,  R )   &    |-  L  =  U_ y  e.  ( G `  p )  pred ( y ,  A ,  R )   &    |-  K  =  U_ y  e.  ( G `  i
 )  pred ( y ,  A ,  R )   &    |-  L  =  U_ y  e.  ( G `  p )  pred ( y ,  A ,  R )   =>    |-  ( ( et  /\  ze )  ->  ( ( G `  m )  =  L  <->  ( G `  suc  i )  =  K ) )
 
Theorembnj556 32912 Technical lemma for bnj852 32933. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( si 
 <->  ( m  e.  D  /\  n  =  suc  m 
 /\  p  e.  m ) )   &    |-  ( et  <->  ( m  e.  D  /\  n  = 
 suc  m  /\  p  e. 
 om  /\  m  =  suc  p ) )   =>    |-  ( et  ->  si )
 
Theorembnj557 32913* Technical lemma for bnj852 32933. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  D  =  ( om  \  { (/)
 } )   &    |-  G  =  ( f  u.  { <. m ,  U_ y  e.  ( f `  p )  pred ( y ,  A ,  R )
 >. } )   &    |-  ( ta  <->  ( f  Fn  m  /\  ph'  /\  ps' ) )   &    |-  ( si  <->  ( m  e.  D  /\  n  = 
 suc  m  /\  p  e.  m ) )   &    |-  ( et 
 <->  ( m  e.  D  /\  n  =  suc  m 
 /\  p  e.  om  /\  m  =  suc  p ) )   &    |-  ( ze  <->  ( i  e. 
 om  /\  suc  i  e.  n  /\  m  = 
 suc  i ) )   &    |-  B  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R )   &    |-  C  =  U_ y  e.  ( f `  p )  pred ( y ,  A ,  R )   &    |-  K  =  U_ y  e.  ( G `  i
 )  pred ( y ,  A ,  R )   &    |-  L  =  U_ y  e.  ( G `  p )  pred ( y ,  A ,  R )   &    |-  G  =  ( f  u.  { <. m ,  C >. } )   &    |-  ( ph'  <->  ( f `  (/) )  =  pred ( x ,  A ,  R ) )   &    |-  ( ps'  <->  A. i  e.  om  ( suc  i  e.  m  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  ( ( R 
 FrSe  A  /\  ta  /\  si )  ->  G  Fn  n )   =>    |-  ( ( R  FrSe  A 
 /\  ta  /\  et  /\  ze )  ->  ( G `  m )  =  L )
 
Theorembnj558 32914* Technical lemma for bnj852 32933. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  D  =  ( om  \  { (/)
 } )   &    |-  G  =  ( f  u.  { <. m ,  U_ y  e.  ( f `  p )  pred ( y ,  A ,  R )
 >. } )   &    |-  ( ta  <->  ( f  Fn  m  /\  ph'  /\  ps' ) )   &    |-  ( si  <->  ( m  e.  D  /\  n  = 
 suc  m  /\  p  e.  m ) )   &    |-  ( et 
 <->  ( m  e.  D  /\  n  =  suc  m 
 /\  p  e.  om  /\  m  =  suc  p ) )   &    |-  ( ze  <->  ( i  e. 
 om  /\  suc  i  e.  n  /\  m  = 
 suc  i ) )   &    |-  B  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R )   &    |-  C  =  U_ y  e.  ( f `  p )  pred ( y ,  A ,  R )   &    |-  K  =  U_ y  e.  ( G `  i
 )  pred ( y ,  A ,  R )   &    |-  L  =  U_ y  e.  ( G `  p )  pred ( y ,  A ,  R )   &    |-  G  =  ( f  u.  { <. m ,  C >. } )   &    |-  ( ph'  <->  ( f `  (/) )  =  pred ( x ,  A ,  R ) )   &    |-  ( ps'  <->  A. i  e.  om  ( suc  i  e.  m  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  ( ( R 
 FrSe  A  /\  ta  /\  si )  ->  G  Fn  n )   =>    |-  ( ( R  FrSe  A 
 /\  ta  /\  et  /\  ze )  ->  ( G ` 
 suc  i )  =  K )
 
Theorembnj561 32915 Technical lemma for bnj852 32933. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( si 
 <->  ( m  e.  D  /\  n  =  suc  m 
 /\  p  e.  m ) )   &    |-  ( et  <->  ( m  e.  D  /\  n  = 
 suc  m  /\  p  e. 
 om  /\  m  =  suc  p ) )   &    |-  (
 ( R  FrSe  A  /\  ta 
 /\  si )  ->  G  Fn  n )   =>    |-  ( ( R  FrSe  A 
 /\  ta  /\  et )  ->  G  Fn  n )
 
Theorembnj562 32916 Technical lemma for bnj852 32933. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( si 
 <->  ( m  e.  D  /\  n  =  suc  m 
 /\  p  e.  m ) )   &    |-  ( et  <->  ( m  e.  D  /\  n  = 
 suc  m  /\  p  e. 
 om  /\  m  =  suc  p ) )   &    |-  (
 ( R  FrSe  A  /\  ta 
 /\  si )  ->  ph" )   =>    |-  ( ( R 
 FrSe  A  /\  ta  /\  et )  ->  ph" )
 
Theorembnj570 32917* Technical lemma for bnj852 32933. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  D  =  ( om  \  { (/)
 } )   &    |-  ( ta  <->  ( f  Fn  m  /\  ph'  /\  ps' ) )   &    |-  ( et  <->  ( m  e.  D  /\  n  = 
 suc  m  /\  p  e. 
 om  /\  m  =  suc  p ) )   &    |-  ( rh 
 <->  ( i  e.  om  /\ 
 suc  i  e.  n  /\  m  =/=  suc  i
 ) )   &    |-  K  =  U_ y  e.  ( G `  i )  pred (
 y ,  A ,  R )   &    |-  G  =  ( f  u.  { <. m ,  C >. } )   &    |-  (
 ( R  FrSe  A  /\  ta 
 /\  et )  ->  G  Fn  n )   &    |-  ( ps'  <->  A. i  e.  om  ( suc  i  e.  m  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   =>    |-  ( ( R  FrSe  A 
 /\  ta  /\  et  /\  rh )  ->  ( G `
  suc  i )  =  K )
 
Theorembnj571 32918* Technical lemma for bnj852 32933. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  D  =  ( om  \  { (/)
 } )   &    |-  G  =  ( f  u.  { <. m ,  U_ y  e.  ( f `  p )  pred ( y ,  A ,  R )
 >. } )   &    |-  ( ta  <->  ( f  Fn  m  /\  ph'  /\  ps' ) )   &    |-  ( si  <->  ( m  e.  D  /\  n  = 
 suc  m  /\  p  e.  m ) )   &    |-  ( et 
 <->  ( m  e.  D  /\  n  =  suc  m 
 /\  p  e.  om  /\  m  =  suc  p ) )   &    |-  ( ze  <->  ( i  e. 
 om  /\  suc  i  e.  n  /\  m  = 
 suc  i ) )   &    |-  B  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R )   &    |-  C  =  U_ y  e.  ( f `  p )  pred ( y ,  A ,  R )   &    |-  K  =  U_ y  e.  ( G `  i
 )  pred ( y ,  A ,  R )   &    |-  L  =  U_ y  e.  ( G `  p )  pred ( y ,  A ,  R )   &    |-  G  =  ( f  u.  { <. m ,  C >. } )   &    |-  ( ph'  <->  ( f `  (/) )  =  pred ( x ,  A ,  R ) )   &    |-  ( ps'  <->  A. i  e.  om  ( suc  i  e.  m  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  ( ( R 
 FrSe  A  /\  ta  /\  si )  ->  G  Fn  n )   &    |-  ( rh  <->  ( i  e. 
 om  /\  suc  i  e.  n  /\  m  =/= 
 suc  i ) )   &    |-  ( ( R  FrSe  A 
 /\  ta  /\  et )  ->  G  Fn  n )   &    |-  (
 ps" 
 <-> 
 A. i  e.  om  ( suc  i  e.  n  ->  ( G `  suc  i )  =  U_ y  e.  ( G `  i
 )  pred ( y ,  A ,  R ) ) )   =>    |-  ( ( R  FrSe  A 
 /\  ta  /\  et )  ->  ps" )
 
Theorembnj605 32919* Technical lemma. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( th 
 <-> 
 A. m  e.  D  ( m  _E  n  -> 
 [. m  /  n ].
 ch ) )   &    |-  ( ph"  <->  [. f  /  f ]. ph )   &    |-  ( ps"  <->  [. f  /  f ]. ps )   &    |-  ( ta  <->  ( f  Fn  m  /\  ph'  /\  ps' ) )   &    |-  ( et  <->  ( m  e.  D  /\  n  = 
 suc  m  /\  p  e. 
 om  /\  m  =  suc  p ) )   &    |-  f  e.  _V   &    |-  ( ch'  <->  ( ( R 
 FrSe  A  /\  x  e.  A )  ->  E! f ( f  Fn  m  /\  ph'  /\  ps' ) ) )   &    |-  ( ph"  <->  ( f `  (/) )  =  pred ( x ,  A ,  R ) )   &    |-  ( ps"  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  ( ( n  =/=  1o  /\  n  e.  D )  ->  E. m E. p et )   &    |-  (
 ( th  /\  m  e.  D  /\  m  _E  n )  ->  ch' )   &    |-  (
 ( R  FrSe  A  /\  ta 
 /\  et )  ->  f  Fn  n )   &    |-  ( ( R 
 FrSe  A  /\  ta  /\  et )  ->  ph" )   &    |-  (
 ( R  FrSe  A  /\  ta 
 /\  et )  ->  ps" )   =>    |-  ( ( n  =/=  1o  /\  n  e.  D  /\  th )  ->  ( ( R  FrSe  A 
 /\  x  e.  A )  ->  E. f ( f  Fn  n  /\  ph  /\  ps ) ) )
 
Theorembnj581 32920* Technical lemma for bnj580 32925. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Unnecessary distinct variable restrictions were removed by Andrew Salmon, 9-Jul-2011.) (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ch 
 <->  ( f  Fn  n  /\  ph  /\  ps )
 )   &    |-  ( ph'  <->  [. g  /  f ]. ph )   &    |-  ( ps'  <->  [. g  /  f ]. ps )   &    |-  ( ch'  <->  [. g  /  f ]. ch )   =>    |-  ( ch'  <->  ( g  Fn  n  /\  ph'  /\  ps' ) )
 
Theorembnj589 32921* Technical lemma for bnj852 32933. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ps 
 <-> 
 A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   =>    |-  ( ps  <->  A. k  e.  om  ( suc  k  e.  n  ->  ( f `  suc  k )  =  U_ y  e.  ( f `  k
 )  pred ( y ,  A ,  R ) ) )
 
Theorembnj590 32922 Technical lemma for bnj852 32933. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ps 
 <-> 
 A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   =>    |-  ( ( B  =  suc  i  /\  ps )  ->  ( i  e.  om  ->  ( B  e.  n  ->  ( f `  B )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) ) )
 
Theorembnj591 32923* Technical lemma for bnj852 32933. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( th 
 <->  ( ( n  e.  D  /\  ch  /\  ch' )  ->  ( f `  j )  =  ( g `  j ) ) )   =>    |-  ( [. k  /  j ]. th  <->  ( ( n  e.  D  /\  ch  /\  ch' )  ->  ( f `  k )  =  ( g `  k ) ) )
 
Theorembnj594 32924* Technical lemma for bnj852 32933. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph 
 <->  ( f `  (/) )  = 
 pred ( x ,  A ,  R )
 )   &    |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  ( ch  <->  ( f  Fn  n  /\  ph  /\  ps ) )   &    |-  D  =  ( om  \  { (/) } )   &    |-  ( ph'  <->  ( g `  (/) )  = 
 pred ( x ,  A ,  R )
 )   &    |-  ( ps'  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( g `  suc  i )  =  U_ y  e.  ( g `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  ( ch'  <->  ( g  Fn  n  /\  ph'  /\  ps' ) )   &    |-  ( th  <->  ( ( n  e.  D  /\  ch  /\  ch' )  ->  ( f `  j )  =  ( g `  j ) ) )   &    |-  ( [. k  /  j ]. th  <->  ( ( n  e.  D  /\  ch  /\  ch' )  ->  ( f `  k )  =  ( g `  k ) ) )   &    |-  ( ta  <->  A. k  e.  n  ( k  _E  j  -> 
 [. k  /  j ]. th ) )   =>    |-  ( ( j  e.  n  /\  ta )  ->  th )
 
Theorembnj580 32925* Technical lemma for bnj579 32926. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph 
 <->  ( f `  (/) )  = 
 pred ( x ,  A ,  R )
 )   &    |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  ( ch  <->  ( f  Fn  n  /\  ph  /\  ps ) )   &    |-  ( ph'  <->  [. g  /  f ]. ph )   &    |-  ( ps'  <->  [. g  /  f ]. ps )   &    |-  ( ch'  <->  [. g  /  f ]. ch )   &    |-  D  =  ( om  \  { (/) } )   &    |-  ( th 
 <->  ( ( n  e.  D  /\  ch  /\  ch' )  ->  ( f `  j )  =  ( g `  j ) ) )   &    |-  ( ta  <->  A. k  e.  n  ( k  _E  j  -> 
 [. k  /  j ]. th ) )   =>    |-  ( n  e.  D  ->  E* f ch )
 
Theorembnj579 32926* Technical lemma for bnj852 32933. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph 
 <->  ( f `  (/) )  = 
 pred ( x ,  A ,  R )
 )   &    |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  D  =  ( om  \  { (/) } )   =>    |-  ( n  e.  D  ->  E* f ( f  Fn  n  /\  ph  /\  ps ) )
 
Theorembnj602 32927 Equality theorem for the  pred function constant. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( X  =  Y  ->  pred
 ( X ,  A ,  R )  =  pred ( Y ,  A ,  R ) )
 
Theorembnj607 32928* Technical lemma for bnj852 32933. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( th 
 <-> 
 A. m  e.  D  ( m  _E  n  -> 
 [. m  /  n ].
 ch ) )   &    |-  ( ph"  <->  [. G  /  f ]. ph )   &    |-  ( ps"  <->  [. G  /  f ]. ps )   &    |-  ( ta  <->  ( f  Fn  m  /\  ph'  /\  ps' ) )   &    |-  ( et  <->  ( m  e.  D  /\  n  = 
 suc  m  /\  p  e. 
 om  /\  m  =  suc  p ) )   &    |-  G  e.  _V   &    |-  ( ch'  <->  ( ( R 
 FrSe  A  /\  x  e.  A )  ->  E! f ( f  Fn  m  /\  ph'  /\  ps' ) ) )   &    |-  ( ph"  <->  ( G `  (/) )  =  pred ( x ,  A ,  R ) )   &    |-  ( ps"  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( G `  suc  i
 )  =  U_ y  e.  ( G `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  ( ( n  =/=  1o  /\  n  e.  D )  ->  E. m E. p et )   &    |-  (
 ( th  /\  m  e.  D  /\  m  _E  n )  ->  ch' )   &    |-  (
 ( R  FrSe  A  /\  ta 
 /\  et )  ->  G  Fn  n )   &    |-  ( ( R 
 FrSe  A  /\  ta  /\  et )  ->  ph" )   &    |-  (
 ( R  FrSe  A  /\  ta 
 /\  et )  ->  ps" )   &    |-  ( ph 
 <->  ( f `  (/) )  = 
 pred ( x ,  A ,  R )
 )   &    |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  ( ph0  <->  [. h  /  f ]. ph )   &    |-  ( ps0  <->  [. h  /  f ]. ps )   &    |-  ( ph1  <->  [. G  /  h ]. ph0 )   &    |-  ( ps1  <->  [. G  /  h ]. ps0 )   =>    |-  ( ( n  =/= 
 1o  /\  n  e.  D  /\  th )  ->  ( ( R  FrSe  A 
 /\  x  e.  A )  ->  E. f ( f  Fn  n  /\  ph  /\  ps ) ) )
 
Theorembnj609 32929* Technical lemma for bnj852 32933. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph 
 <->  ( f `  (/) )  = 
 pred ( X ,  A ,  R )
 )   &    |-  ( ph"  <->  [. G  /  f ]. ph )   &    |-  G  e.  _V   =>    |-  ( ph"  <->  ( G `  (/) )  =  pred ( X ,  A ,  R ) )
 
Theorembnj611 32930* Technical lemma for bnj852 32933. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ps 
 <-> 
 A. i  e.  om  ( suc  i  e.  N  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  ( ps"  <->  [. G  /  f ]. ps )   &    |-  G  e.  _V   =>    |-  ( ps"  <->  A. i  e.  om  ( suc  i  e.  N  ->  ( G `  suc  i
 )  =  U_ y  e.  ( G `  i
 )  pred ( y ,  A ,  R ) ) )
 
Theorembnj600 32931* Technical lemma for bnj852 32933. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph 
 <->  ( f `  (/) )  = 
 pred ( x ,  A ,  R )
 )   &    |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  D  =  ( om  \  { (/) } )   &    |-  ( ch 
 <->  ( ( R  FrSe  A 
 /\  x  e.  A )  ->  E! f ( f  Fn  n  /\  ph 
 /\  ps ) ) )   &    |-  ( th  <->  A. m  e.  D  ( m  _E  n  -> 
 [. m  /  n ].
 ch ) )   &    |-  ( ph'  <->  [. m  /  n ]. ph )   &    |-  ( ps'  <->  [. m  /  n ].
 ps )   &    |-  ( ch'  <->  [. m  /  n ].
 ch )   &    |-  ( ph"  <->  [. G  /  f ]. ph )   &    |-  ( ps"  <->  [. G  /  f ]. ps )   &    |-  ( ch"  <->  [. G  /  f ]. ch )   &    |-  G  =  ( f  u.  { <. m ,  U_ y  e.  ( f `  p )  pred ( y ,  A ,  R )
 >. } )   &    |-  ( ta  <->  ( f  Fn  m  /\  ph'  /\  ps' ) )   &    |-  ( si  <->  ( m  e.  D  /\  n  = 
 suc  m  /\  p  e.  m ) )   &    |-  ( et 
 <->  ( m  e.  D  /\  n  =  suc  m 
 /\  p  e.  om  /\  m  =  suc  p ) )   &    |-  ( ze  <->  ( i  e. 
 om  /\  suc  i  e.  n  /\  m  = 
 suc  i ) )   &    |-  ( rh  <->  ( i  e. 
 om  /\  suc  i  e.  n  /\  m  =/= 
 suc  i ) )   &    |-  B  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R )   &    |-  C  =  U_ y  e.  ( f `  p )  pred ( y ,  A ,  R )   &    |-  K  =  U_ y  e.  ( G `  i
 )  pred ( y ,  A ,  R )   &    |-  L  =  U_ y  e.  ( G `  p )  pred ( y ,  A ,  R )   &    |-  G  =  ( f  u.  { <. m ,  C >. } )   =>    |-  ( n  =/=  1o  ->  ( ( n  e.  D  /\  th )  ->  ch ) )
 
Theorembnj601 32932* Technical lemma for bnj852 32933. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph 
 <->  ( f `  (/) )  = 
 pred ( x ,  A ,  R )
 )   &    |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  D  =  ( om  \  { (/) } )   &    |-  ( ch 
 <->  ( ( R  FrSe  A 
 /\  x  e.  A )  ->  E! f ( f  Fn  n  /\  ph 
 /\  ps ) ) )   &    |-  ( th  <->  A. m  e.  D  ( m  _E  n  -> 
 [. m  /  n ].
 ch ) )   =>    |-  ( n  =/= 
 1o  ->  ( ( n  e.  D  /\  th )  ->  ch ) )
 
Theorembnj852 32933* Technical lemma for bnj69 33020. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph 
 <->  ( f `  (/) )  = 
 pred ( X ,  A ,  R )
 )   &    |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  D  =  ( om  \  { (/) } )   =>    |-  (
 ( R  FrSe  A  /\  X  e.  A )  ->  A. n  e.  D  E! f ( f  Fn  n  /\  ph  /\  ps ) )
 
Theorembnj864 32934* Technical lemma for bnj69 33020. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph 
 <->  ( f `  (/) )  = 
 pred ( X ,  A ,  R )
 )   &    |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  D  =  ( om  \  { (/) } )   &    |-  ( ch 
 <->  ( R  FrSe  A  /\  X  e.  A  /\  n  e.  D )
 )   &    |-  ( th  <->  ( f  Fn  n  /\  ph  /\  ps ) )   =>    |-  ( ch  ->  E! f th )
 
Theorembnj865 32935* Technical lemma for bnj69 33020. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph 
 <->  ( f `  (/) )  = 
 pred ( X ,  A ,  R )
 )   &    |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  D  =  ( om  \  { (/) } )   &    |-  ( ch 
 <->  ( R  FrSe  A  /\  X  e.  A  /\  n  e.  D )
 )   &    |-  ( th  <->  ( f  Fn  n  /\  ph  /\  ps ) )   =>    |- 
 E. w A. n ( ch  ->  E. f  e.  w  th )
 
Theorembnj873 32936* Technical lemma for bnj69 33020. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  B  =  { f  |  E. n  e.  D  (
 f  Fn  n  /\  ph 
 /\  ps ) }   &    |-  ( ph'  <->  [. g  /  f ]. ph )   &    |-  ( ps'  <->  [. g  /  f ]. ps )   =>    |-  B  =  { g  |  E. n  e.  D  ( g  Fn  n  /\  ph'  /\  ps' ) }
 
Theorembnj849 32937* Technical lemma for bnj69 33020. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Mario Carneiro, 22-Dec-2016.) (New usage is discouraged.)
 |-  ( ph 
 <->  ( f `  (/) )  = 
 pred ( X ,  A ,  R )
 )   &    |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  D  =  ( om  \  { (/) } )   &    |-  B  =  { f  |  E. n  e.  D  (
 f  Fn  n  /\  ph 
 /\  ps ) }   &    |-  ( ch 
 <->  ( R  FrSe  A  /\  X  e.  A  /\  n  e.  D )
 )   &    |-  ( th  <->  ( f  Fn  n  /\  ph  /\  ps ) )   &    |-  ( ph'  <->  [. g  /  f ]. ph )   &    |-  ( ps'  <->  [. g  /  f ]. ps )   &    |-  ( th'  <->  [. g  /  f ]. th )   &    |-  ( ta  <->  ( R  FrSe  A 
 /\  X  e.  A ) )   =>    |-  ( ( R  FrSe  A 
 /\  X  e.  A )  ->  B  e.  _V )
 
Theorembnj882 32938* Definition (using hypotheses for readability) of the function giving the transitive closure of  X in  A by  R. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph 
 <->  ( f `  (/) )  = 
 pred ( X ,  A ,  R )
 )   &    |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  D  =  ( om  \  { (/) } )   &    |-  B  =  { f  |  E. n  e.  D  (
 f  Fn  n  /\  ph 
 /\  ps ) }   =>    |-  trCl ( X ,  A ,  R )  =  U_ f  e.  B  U_ i  e.  dom  f
 ( f `  i
 )
 
Theorembnj18eq1 32939 Equality theorem for transitive closure. (Contributed by Mario Carneiro, 22-Dec-2016.) (New usage is discouraged.)
 |-  ( X  =  Y  ->  trCl
 ( X ,  A ,  R )  =  trCl ( Y ,  A ,  R ) )
 
Theorembnj893 32940 Property of  trCl. Under certain conditions, the transitive closure of  X in  A by  R is a set. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  (
 ( R  FrSe  A  /\  X  e.  A )  -> 
 trCl ( X ,  A ,  R )  e.  _V )
 
Theorembnj900 32941* Technical lemma for bnj69 33020. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  D  =  ( om  \  { (/)
 } )   &    |-  B  =  {
 f  |  E. n  e.  D  ( f  Fn  n  /\  ph  /\  ps ) }   =>    |-  ( f  e.  B  -> 
 (/)  e.  dom  f )
 
Theorembnj906 32942 Property of  trCl. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  (
 ( R  FrSe  A  /\  X  e.  A )  -> 
 pred ( X ,  A ,  R )  C_ 
 trCl ( X ,  A ,  R )
 )
 
Theorembnj908 32943* Technical lemma for bnj69 33020. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph 
 <->  ( f `  (/) )  = 
 pred ( x ,  A ,  R )
 )   &    |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  D  =  ( om  \  { (/) } )   &    |-  ( ch 
 <->  ( ( R  FrSe  A 
 /\  x  e.  A )  ->  E! f ( f  Fn  n  /\  ph 
 /\  ps ) ) )   &    |-  ( th  <->  A. m  e.  D  ( m  _E  n  -> 
 [. m  /  n ].
 ch ) )   &    |-  ( ph'  <->  [. m  /  n ]. ph )   &    |-  ( ps'  <->  [. m  /  n ].
 ps )   &    |-  ( ch'  <->  [. m  /  n ].
 ch )   &    |-  ( ph"  <->  [. G  /  f ]. ph )   &    |-  ( ps"  <->  [. G  /  f ]. ps )   &    |-  ( ch"  <->  [. G  /  f ]. ch )   &    |-  G  =  ( f  u.  { <. m ,  U_ y  e.  ( f `  p )  pred ( y ,  A ,  R )
 >. } )   &    |-  ( ta  <->  ( f  Fn  m  /\  ph'  /\  ps' ) )   &    |-  ( si  <->  ( m  e.  D  /\  n  = 
 suc  m  /\  p  e.  m ) )   &    |-  ( et 
 <->  ( m  e.  D  /\  n  =  suc  m 
 /\  p  e.  om  /\  m  =  suc  p ) )   &    |-  ( ze  <->  ( i  e. 
 om  /\  suc  i  e.  n  /\  m  = 
 suc  i ) )   &    |-  ( rh  <->  ( i  e. 
 om  /\  suc  i  e.  n  /\  m  =/= 
 suc  i ) )   &    |-  B  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R )   &    |-  C  =  U_ y  e.  ( f `  p )  pred ( y ,  A ,  R )   &    |-  K  =  U_ y  e.  ( G `  i
 )  pred ( y ,  A ,  R )   &    |-  L  =  U_ y  e.  ( G `  p )  pred ( y ,  A ,  R )   &    |-  G  =  ( f  u.  { <. m ,  C >. } )   =>    |-  ( ( R  FrSe  A 
 /\  x  e.  A  /\  ch'  /\  et )  ->  E. f ( G  Fn  n  /\  ph"  /\  ps" ) )
 
Theorembnj911 32944* Technical lemma for bnj69 33020. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph 
 <->  ( f `  (/) )  = 
 pred ( X ,  A ,  R )
 )   &    |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   =>    |-  ( ( f  Fn  n  /\  ph  /\  ps )  ->  A. i ( f  Fn  n  /\  ph  /\  ps ) )
 
Theorembnj916 32945* Technical lemma for bnj69 33020. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph 
 <->  ( f `  (/) )  = 
 pred ( X ,  A ,  R )
 )   &    |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  D  =  ( om  \  { (/) } )   &    |-  B  =  { f  |  E. n  e.  D  (
 f  Fn  n  /\  ph 
 /\  ps ) }   &    |-  ( ch 
 <->  ( f  Fn  n  /\  ph  /\  ps )
 )   =>    |-  ( y  e.  trCl ( X ,  A ,  R )  ->  E. f E. n E. i ( n  e.  D  /\  ch 
 /\  i  e.  n  /\  y  e.  (
 f `  i )
 ) )
 
Theorembnj917 32946* Technical lemma for bnj69 33020. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph 
 <->  ( f `  (/) )  = 
 pred ( X ,  A ,  R )
 )   &    |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  D  =  ( om  \  { (/) } )   &    |-  B  =  { f  |  E. n  e.  D  (
 f  Fn  n  /\  ph 
 /\  ps ) }   &    |-  ( ch 
 <->  ( n  e.  D  /\  f  Fn  n  /\  ph  /\  ps )
 )   =>    |-  ( y  e.  trCl ( X ,  A ,  R )  ->  E. f E. n E. i ( ch  /\  i  e.  n  /\  y  e.  ( f `  i
 ) ) )
 
Theorembnj934 32947* Technical lemma for bnj69 33020. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph 
 <->  ( f `  (/) )  = 
 pred ( X ,  A ,  R )
 )   &    |-  ( ph'  <->  [. p  /  n ].
 ph )   &    |-  ( ph"  <->  [. G  /  f ]. ph' )   &    |-  G  e.  _V   =>    |-  (
 ( ph  /\  ( G `
  (/) )  =  ( f `  (/) ) ) 
 ->  ph" )
 
Theorembnj929 32948* Technical lemma for bnj69 33020. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph 
 <->  ( f `  (/) )  = 
 pred ( X ,  A ,  R )
 )   &    |-  ( ph'  <->  [. p  /  n ].
 ph )   &    |-  ( ph"  <->  [. G  /  f ]. ph' )   &    |-  D  =  ( om  \  { (/) } )   &    |-  G  =  ( f  u.  { <. n ,  C >. } )   &    |-  C  e.  _V   =>    |-  (
 ( n  e.  D  /\  p  =  suc  n 
 /\  f  Fn  n  /\  ph )  ->  ph" )
 
Theorembnj938 32949* Technical lemma for bnj69 33020. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  D  =  ( om  \  { (/)
 } )   &    |-  ( ta  <->  ( f  Fn  m  /\  ph'  /\  ps' ) )   &    |-  ( si  <->  ( m  e.  D  /\  n  = 
 suc  m  /\  p  e.  m ) )   &    |-  ( ph'  <->  ( f `  (/) )  = 
 pred ( X ,  A ,  R )
 )   &    |-  ( ps'  <->  A. i  e.  om  ( suc  i  e.  m  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   =>    |-  ( ( R  FrSe  A 
 /\  X  e.  A  /\  ta  /\  si )  -> 
 U_ y  e.  (
 f `  p )  pred ( y ,  A ,  R )  e.  _V )
 
Theorembnj944 32950* Technical lemma for bnj69 33020. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph 
 <->  ( f `  (/) )  = 
 pred ( X ,  A ,  R )
 )   &    |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph  /\  ps ) )   &    |-  ( ph'  <->  [. p  /  n ].
 ph )   &    |-  ( ph"  <->  [. G  /  f ]. ph' )   &    |-  D  =  ( om  \  { (/) } )   &    |-  C  =  U_ y  e.  (
 f `  m )  pred ( y ,  A ,  R )   &    |-  G  =  ( f  u.  { <. n ,  C >. } )   &    |-  ( ta 
 <->  ( f  Fn  n  /\  ph  /\  ps )
 )   &    |-  ( si  <->  ( n  e.  D  /\  p  = 
 suc  n  /\  m  e.  n ) )   =>    |-  ( ( ( R  FrSe  A  /\  X  e.  A )  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) ) 
 ->  ph" )
 
Theorembnj953 32951 Technical lemma for bnj69 33020. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ps 
 <-> 
 A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  ( ( G `
  i )  =  ( f `  i
 )  ->  A. y ( G `  i )  =  ( f `  i ) )   =>    |-  ( ( ( G `  i )  =  ( f `  i )  /\  ( G `
  suc  i )  =  ( f `  suc  i )  /\  ( i  e.  om  /\  suc  i  e.  n )  /\  ps )  ->  ( G `  suc  i )  =  U_ y  e.  ( G `  i
 )  pred ( y ,  A ,  R ) )
 
Theorembnj958 32952* Technical lemma for bnj69 33020. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  C  =  U_ y  e.  (
 f `  m )  pred ( y ,  A ,  R )   &    |-  G  =  ( f  u.  { <. n ,  C >. } )   =>    |-  (
 ( G `  i
 )  =  ( f `
  i )  ->  A. y ( G `  i )  =  (
 f `  i )
 )
 
Theorembnj1000 32953* Technical lemma for bnj852 32933. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ps 
 <-> 
 A. i  e.  om  ( suc  i  e.  N  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  ( ps"  <->  [. G  /  f ]. ps )   &    |-  G  e.  _V   &    |-  C  =  U_ y  e.  (
 f `  m )  pred ( y ,  A ,  R )   &    |-  G  =  ( f  u.  { <. n ,  C >. } )   =>    |-  ( ps"  <->  A. i  e.  om  ( suc  i  e.  N  ->  ( G `  suc  i
 )  =  U_ y  e.  ( G `  i
 )  pred ( y ,  A ,  R ) ) )
 
Theorembnj965 32954* Technical lemma for bnj852 32933. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ps 
 <-> 
 A. i  e.  om  ( suc  i  e.  N  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  ( ps"  <->  [. G  /  f ]. ps )   &    |-  C  =  U_ y  e.  ( f `  m )  pred (
 y ,  A ,  R )   &    |-  G  =  ( f  u.  { <. n ,  C >. } )   =>    |-  ( ps"  <->  A. i  e.  om  ( suc  i  e.  N  ->  ( G `  suc  i
 )  =  U_ y  e.  ( G `  i
 )  pred ( y ,  A ,  R ) ) )
 
Theorembnj964 32955* Technical lemma for bnj69 33020. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ps 
 <-> 
 A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph  /\  ps ) )   &    |-  ( ps'  <->  [. p  /  n ].
 ps )   &    |-  ( ps"  <->  [. G  /  f ]. ps' )   &    |-  C  =  U_ y  e.  ( f `  m )  pred (
 y ,  A ,  R )   &    |-  G  =  ( f  u.  { <. n ,  C >. } )   &    |-  (
 ( ( R  FrSe  A 
 /\  X  e.  A )  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) 
 /\  ( i  e. 
 om  /\  suc  i  e.  p  /\  suc  i  e.  n ) )  ->  ( G `  suc  i
 )  =  U_ y  e.  ( G `  i
 )  pred ( y ,  A ,  R ) )   &    |-  ( ( ( R  FrSe  A  /\  X  e.  A )  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n )  /\  ( i  e.  om  /\  suc  i  e.  p  /\  n  =  suc  i ) )  ->  ( G ` 
 suc  i )  = 
 U_ y  e.  ( G `  i )  pred ( y ,  A ,  R ) )   =>    |-  ( ( ( R  FrSe  A  /\  X  e.  A )  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) ) 
 ->  ps" )
 
Theorembnj966 32956* Technical lemma for bnj69 33020. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ch 
 <->  ( n  e.  D  /\  f  Fn  n  /\  ph  /\  ps )
 )   &    |-  D  =  ( om  \  { (/) } )   &    |-  C  =  U_ y  e.  (
 f `  m )  pred ( y ,  A ,  R )   &    |-  G  =  ( f  u.  { <. n ,  C >. } )   &    |-  (
 ( ( R  FrSe  A 
 /\  X  e.  A )  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) )  ->  C  e.  _V )   &    |-  ( ( ( R  FrSe  A  /\  X  e.  A )  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) ) 
 ->  G  Fn  p )   =>    |-  ( ( ( R 
 FrSe  A  /\  X  e.  A )  /\  ( ch 
 /\  n  =  suc  m 
 /\  p  =  suc  n )  /\  ( i  e.  om  /\  suc  i  e.  p  /\  n  =  suc  i ) )  ->  ( G ` 
 suc  i )  = 
 U_ y  e.  ( G `  i )  pred ( y ,  A ,  R ) )
 
Theorembnj967 32957* Technical lemma for bnj69 33020. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ps 
 <-> 
 A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph  /\  ps ) )   &    |-  D  =  ( om  \  { (/) } )   &    |-  C  =  U_ y  e.  (
 f `  m )  pred ( y ,  A ,  R )   &    |-  G  =  ( f  u.  { <. n ,  C >. } )   &    |-  (
 ( ( R  FrSe  A 
 /\  X  e.  A )  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) )  ->  C  e.  _V )   =>    |-  ( ( ( R 
 FrSe  A  /\  X  e.  A )  /\  ( ch 
 /\  n  =  suc  m 
 /\  p  =  suc  n )  /\  ( i  e.  om  /\  suc  i  e.  p  /\  suc  i  e.  n ) )  ->  ( G ` 
 suc  i )  = 
 U_ y  e.  ( G `  i )  pred ( y ,  A ,  R ) )
 
Theorembnj969 32958* Technical lemma for bnj69 33020. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph 
 <->  ( f `  (/) )  = 
 pred ( X ,  A ,  R )
 )   &    |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph  /\  ps ) )   &    |-  D  =  ( om  \  { (/) } )   &    |-  C  =  U_ y  e.  (
 f `  m )  pred ( y ,  A ,  R )   &    |-  ( ta  <->  ( f  Fn  n  /\  ph  /\  ps ) )   &    |-  ( si  <->  ( n  e.  D  /\  p  = 
 suc  n  /\  m  e.  n ) )   =>    |-  ( ( ( R  FrSe  A  /\  X  e.  A )  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) ) 
 ->  C  e.  _V )
 
Theorembnj970 32959 Technical lemma for bnj69 33020. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ch 
 <->  ( n  e.  D  /\  f  Fn  n  /\  ph  /\  ps )
 )   &    |-  D  =  ( om  \  { (/) } )   =>    |-  ( ( ( R  FrSe  A  /\  X  e.  A )  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) ) 
 ->  p  e.  D )
 
Theorembnj910 32960* Technical lemma for bnj69 33020. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph 
 <->  ( f `  (/) )  = 
 pred ( X ,  A ,  R )
 )   &    |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph  /\  ps ) )   &    |-  ( ph'  <->  [. p  /  n ].
 ph )   &    |-  ( ps'  <->  [. p  /  n ].
 ps )   &    |-  ( ch'  <->  [. p  /  n ].
 ch )   &    |-  ( ph"  <->  [. G  /  f ]. ph' )   &    |-  ( ps"  <->  [. G  /  f ]. ps' )   &    |-  ( ch"  <->  [. G  /  f ]. ch' )   &    |-  D  =  ( om  \  { (/) } )   &    |-  B  =  { f  |  E. n  e.  D  (
 f  Fn  n  /\  ph 
 /\  ps ) }   &    |-  C  =  U_ y  e.  (
 f `  m )  pred ( y ,  A ,  R )   &    |-  G  =  ( f  u.  { <. n ,  C >. } )   &    |-  ( ta 
 <->  ( f  Fn  n  /\  ph  /\  ps )
 )   &    |-  ( si  <->  ( n  e.  D  /\  p  = 
 suc  n  /\  m  e.  n ) )   =>    |-  ( ( ( R  FrSe  A  /\  X  e.  A )  /\  ( ch  /\  n  =  suc  m  /\  p  =  suc  n ) ) 
 ->  ch" )
 
Theorembnj978 32961* Technical lemma for bnj69 33020. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( th 
 <->  ( R  FrSe  A  /\  X  e.  A  /\  y  e.  trCl ( X ,  A ,  R )  /\  z  e.  pred ( y ,  A ,  R ) ) )   &    |-  ( th  ->  z  e.  trCl
 ( X ,  A ,  R ) )   =>    |-  ( ( R 
 FrSe  A  /\  X  e.  A )  ->  TrFo (  trCl ( X ,  A ,  R ) ,  A ,  R ) )
 
Theorembnj981 32962* Technical lemma for bnj69 33020. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph 
 <->  ( f `  (/) )  = 
 pred ( X ,  A ,  R )
 )   &    |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  D  =  ( om  \  { (/) } )   &    |-  B  =  { f  |  E. n  e.  D  (
 f  Fn  n  /\  ph 
 /\  ps ) }   &    |-  ( ch 
 <->  ( n  e.  D  /\  f  Fn  n  /\  ph  /\  ps )
 )   =>    |-  ( Z  e.  trCl ( X ,  A ,  R )  ->  E. f E. n E. i ( ch  /\  i  e.  n  /\  Z  e.  ( f `  i
 ) ) )
 
Theorembnj983 32963* Technical lemma for bnj69 33020. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph 
 <->  ( f `  (/) )  = 
 pred ( X ,  A ,  R )
 )   &    |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  D  =  ( om  \  { (/) } )   &    |-  B  =  { f  |  E. n  e.  D  (
 f  Fn  n  /\  ph 
 /\  ps ) }   &    |-  ( ch 
 <->  ( n  e.  D  /\  f  Fn  n  /\  ph  /\  ps )
 )   =>    |-  ( Z  e.  trCl ( X ,  A ,  R )  <->  E. f E. n E. i ( ch  /\  i  e.  n  /\  Z  e.  ( f `  i ) ) )
 
Theorembnj984 32964 Technical lemma for bnj69 33020. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ch 
 <->  ( n  e.  D  /\  f  Fn  n  /\  ph  /\  ps )
 )   &    |-  B  =  { f  |  E. n  e.  D  ( f  Fn  n  /\  ph  /\  ps ) }   =>    |-  ( G  e.  A  ->  ( G  e.  B  <->  [. G  /  f ]. E. n ch ) )
 
Theorembnj985 32965* Technical lemma for bnj69 33020. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ch 
 <->  ( n  e.  D  /\  f  Fn  n  /\  ph  /\  ps )
 )   &    |-  ( ch'  <->  [. p  /  n ].
 ch )   &    |-  ( ch"  <->  [. G  /  f ]. ch' )   &    |-  B  =  {
 f  |  E. n  e.  D  ( f  Fn  n  /\  ph  /\  ps ) }   &    |-  G  =  ( f  u.  { <. n ,  C >. } )   =>    |-  ( G  e.  B  <->  E. p ch" )
 
Theorembnj986 32966* Technical lemma for bnj69 33020. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ch 
 <->  ( n  e.  D  /\  f  Fn  n  /\  ph  /\  ps )
 )   &    |-  D  =  ( om  \  { (/) } )   &    |-  ( ta 
 <->  ( m  e.  om  /\  n  =  suc  m  /\  p  =  suc  n ) )   =>    |-  ( ch  ->  E. m E. p ta )
 
Theorembnj996 32967* Technical lemma for bnj69 33020. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph 
 <->  ( f `  (/) )  = 
 pred ( X ,  A ,  R )
 )   &    |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph  /\  ps ) )   &    |-  ( th  <->  ( R  FrSe  A 
 /\  X  e.  A  /\  y  e.  trCl ( X ,  A ,  R )  /\  z  e. 
 pred ( y ,  A ,  R ) ) )   &    |-  ( ta  <->  ( m  e. 
 om  /\  n  =  suc  m  /\  p  = 
 suc  n ) )   &    |-  ( et  <->  ( i  e.  n  /\  y  e.  ( f `  i
 ) ) )   &    |-  D  =  ( om  \  { (/)
 } )   &    |-  B  =  {
 f  |  E. n  e.  D  ( f  Fn  n  /\  ph  /\  ps ) }   =>    |- 
 E. f E. n E. i E. m E. p ( th  ->  ( ch  /\  ta  /\  et ) )
 
Theorembnj998 32968* Technical lemma for bnj69 33020. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph 
 <->  ( f `  (/) )  = 
 pred ( X ,  A ,  R )
 )   &    |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph  /\  ps ) )   &    |-  ( th  <->  ( R  FrSe  A 
 /\  X  e.  A  /\  y  e.  trCl ( X ,  A ,  R )  /\  z  e. 
 pred ( y ,  A ,  R ) ) )   &    |-  ( ta  <->  ( m  e. 
 om  /\  n  =  suc  m  /\  p  = 
 suc  n ) )   &    |-  ( ph'  <->  [. p  /  n ].
 ph )   &    |-  ( ps'  <->  [. p  /  n ].
 ps )   &    |-  ( ch'  <->  [. p  /  n ].
 ch )   &    |-  ( ph"  <->  [. G  /  f ]. ph' )   &    |-  ( ps"  <->  [. G  /  f ]. ps' )   &    |-  ( ch"  <->  [. G  /  f ]. ch' )   &    |-  D  =  ( om  \  { (/) } )   &    |-  B  =  { f  |  E. n  e.  D  (
 f  Fn  n  /\  ph 
 /\  ps ) }   &    |-  C  =  U_ y  e.  (
 f `  m )  pred ( y ,  A ,  R )   &    |-  G  =  ( f  u.  { <. n ,  C >. } )   =>    |-  (
 ( th  /\  ch  /\  ta 
 /\  et )  ->  ch" )
 
Theorembnj999 32969* Technical lemma for bnj69 33020. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph 
 <->  ( f `  (/) )  = 
 pred ( X ,  A ,  R )
 )   &    |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph  /\  ps ) )   &    |-  ( ph'  <->  [. p  /  n ].
 ph )   &    |-  ( ps'  <->  [. p  /  n ].
 ps )   &    |-  ( ch'  <->  [. p  /  n ].
 ch )   &    |-  ( ph"  <->  [. G  /  f ]. ph' )   &    |-  ( ps"  <->  [. G  /  f ]. ps' )   &    |-  ( ch"  <->  [. G  /  f ]. ch' )   &    |-  C  =  U_ y  e.  ( f `  m )  pred (
 y ,  A ,  R )   &    |-  G  =  ( f  u.  { <. n ,  C >. } )   =>    |-  (
 ( ch"  /\  i  e.  om 
 /\  suc  i  e.  p  /\  y  e.  ( G `  i ) ) 
 ->  pred ( y ,  A ,  R ) 
 C_  ( G `  suc  i ) )
 
Theorembnj1001 32970 Technical lemma for bnj69 33020. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ch 
 <->  ( n  e.  D  /\  f  Fn  n  /\  ph  /\  ps )
 )   &    |-  ( ta  <->  ( m  e. 
 om  /\  n  =  suc  m  /\  p  = 
 suc  n ) )   &    |-  ( et  <->  ( i  e.  n  /\  y  e.  ( f `  i
 ) ) )   &    |-  D  =  ( om  \  { (/)
 } )   &    |-  ( ( th  /\ 
 ch  /\  ta  /\  et )  ->  ch" )   =>    |-  ( ( th  /\  ch 
 /\  ta  /\  et )  ->  ( ch"  /\  i  e.  om 
 /\  suc  i  e.  p ) )
 
Theorembnj1006 32971* Technical lemma for bnj69 33020. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph 
 <->  ( f `  (/) )  = 
 pred ( X ,  A ,  R )
 )   &    |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph  /\  ps ) )   &    |-  ( th  <->  ( R  FrSe  A 
 /\  X  e.  A  /\  y  e.  trCl ( X ,  A ,  R )  /\  z  e. 
 pred ( y ,  A ,  R ) ) )   &    |-  ( ta  <->  ( m  e. 
 om  /\  n  =  suc  m  /\  p  = 
 suc  n ) )   &    |-  ( et  <->  ( i  e.  n  /\  y  e.  ( f `  i
 ) ) )   &    |-  ( ph'  <->  [. p  /  n ]. ph )   &    |-  ( ps'  <->  [. p  /  n ].
 ps )   &    |-  ( ch'  <->  [. p  /  n ].
 ch )   &    |-  ( ph"  <->  [. G  /  f ]. ph' )   &    |-  ( ps"  <->  [. G  /  f ]. ps' )   &    |-  ( ch"  <->  [. G  /  f ]. ch' )   &    |-  D  =  ( om  \  { (/) } )   &    |-  C  =  U_ y  e.  (
 f `  m )  pred ( y ,  A ,  R )   &    |-  G  =  ( f  u.  { <. n ,  C >. } )   &    |-  (
 ( th  /\  ch  /\  ta 
 /\  et )  ->  ( ch"  /\  i  e.  om  /\  suc  i  e.  p ) )   =>    |-  ( ( th  /\  ch 
 /\  ta  /\  et )  -> 
 pred ( y ,  A ,  R ) 
 C_  ( G `  suc  i ) )
 
Theorembnj1014 32972* Technical lemma for bnj69 33020. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph 
 <->  ( f `  (/) )  = 
 pred ( X ,  A ,  R )
 )   &    |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  D  =  ( om  \  { (/) } )   &    |-  B  =  { f  |  E. n  e.  D  (
 f  Fn  n  /\  ph 
 /\  ps ) }   =>    |-  ( ( g  e.  B  /\  j  e.  dom  g )  ->  ( g `  j
 )  C_  trCl ( X ,  A ,  R ) )
 
Theorembnj1015 32973* Technical lemma for bnj69 33020. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph 
 <->  ( f `  (/) )  = 
 pred ( X ,  A ,  R )
 )   &    |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  D  =  ( om  \  { (/) } )   &    |-  B  =  { f  |  E. n  e.  D  (
 f  Fn  n  /\  ph 
 /\  ps ) }   &    |-  G  e.  V   &    |-  J  e.  V   =>    |-  (
 ( G  e.  B  /\  J  e.  dom  G )  ->  ( G `  J )  C_  trCl ( X ,  A ,  R ) )
 
Theorembnj1018 32974* Technical lemma for bnj69 33020. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph 
 <->  ( f `  (/) )  = 
 pred ( X ,  A ,  R )
 )   &    |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph  /\  ps ) )   &    |-  ( th  <->  ( R  FrSe  A 
 /\  X  e.  A  /\  y  e.  trCl ( X ,  A ,  R )  /\  z  e. 
 pred ( y ,  A ,  R ) ) )   &    |-  ( ta  <->  ( m  e. 
 om  /\  n  =  suc  m  /\  p  = 
 suc  n ) )   &    |-  ( ph'  <->  [. p  /  n ].
 ph )   &    |-  ( ps'  <->  [. p  /  n ].
 ps )   &    |-  ( ch'  <->  [. p  /  n ].
 ch )   &    |-  ( ph"  <->  [. G  /  f ]. ph' )   &    |-  ( ps"  <->  [. G  /  f ]. ps' )   &    |-  ( ch"  <->  [. G  /  f ]. ch' )   &    |-  D  =  ( om  \  { (/) } )   &    |-  B  =  { f  |  E. n  e.  D  (
 f  Fn  n  /\  ph 
 /\  ps ) }   &    |-  C  =  U_ y  e.  (
 f `  m )  pred ( y ,  A ,  R )   &    |-  G  =  ( f  u.  { <. n ,  C >. } )   &    |-  ( ch"  <->  ( p  e.  D  /\  G  Fn  p  /\  ph"  /\  ps" ) )   &    |-  ( ( th  /\  ch 
 /\  ta  /\  et )  ->  ch" )   &    |-  ( ( th  /\ 
 ch  /\  ta  /\  et )  ->  ( ch"  /\  i  e.  om  /\  suc  i  e.  p ) )   =>    |-  ( ( th  /\ 
 ch  /\  et  /\  E. p ta )  ->  ( G `  suc  i ) 
 C_  trCl ( X ,  A ,  R )
 )
 
Theorembnj1020 32975* Technical lemma for bnj69 33020. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph 
 <->  ( f `  (/) )  = 
 pred ( X ,  A ,  R )
 )   &    |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph  /\  ps ) )   &    |-  ( th  <->  ( R  FrSe  A 
 /\  X  e.  A  /\  y  e.  trCl ( X ,  A ,  R )  /\  z  e. 
 pred ( y ,  A ,  R ) ) )   &    |-  ( ta  <->  ( m  e. 
 om  /\  n  =  suc  m  /\  p  = 
 suc  n ) )   &    |-  ( et  <->  ( i  e.  n  /\  y  e.  ( f `  i
 ) ) )   &    |-  ( ph'  <->  [. p  /  n ]. ph )   &    |-  ( ps'  <->  [. p  /  n ].
 ps )   &    |-  ( ch'  <->  [. p  /  n ].
 ch )   &    |-  ( ph"  <->  [. G  /  f ]. ph' )   &    |-  ( ps"  <->  [. G  /  f ]. ps' )   &    |-  ( ch"  <->  [. G  /  f ]. ch' )   &    |-  D  =  ( om  \  { (/) } )   &    |-  B  =  { f  |  E. n  e.  D  (
 f  Fn  n  /\  ph 
 /\  ps ) }   &    |-  C  =  U_ y  e.  (
 f `  m )  pred ( y ,  A ,  R )   &    |-  G  =  ( f  u.  { <. n ,  C >. } )   &    |-  ( ch"  <->  ( p  e.  D  /\  G  Fn  p  /\  ph"  /\  ps" ) )   =>    |-  ( ( th  /\  ch 
 /\  et  /\  E. p ta )  ->  pred (
 y ,  A ,  R )  C_  trCl ( X ,  A ,  R ) )
 
Theorembnj1021 32976* Technical lemma for bnj69 33020. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph 
 <->  ( f `  (/) )  = 
 pred ( X ,  A ,  R )
 )   &    |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph  /\  ps ) )   &    |-  ( th  <->  ( R  FrSe  A 
 /\  X  e.  A  /\  y  e.  trCl ( X ,  A ,  R )  /\  z  e. 
 pred ( y ,  A ,  R ) ) )   &    |-  ( ta  <->  ( m  e. 
 om  /\  n  =  suc  m  /\  p  = 
 suc  n ) )   &    |-  ( et  <->  ( i  e.  n  /\  y  e.  ( f `  i
 ) ) )   &    |-  D  =  ( om  \  { (/)
 } )   &    |-  B  =  {
 f  |  E. n  e.  D  ( f  Fn  n  /\  ph  /\  ps ) }   =>    |- 
 E. f E. n E. i E. m ( th  ->  ( th  /\ 
 ch  /\  et  /\  E. p ta ) )
 
Theorembnj907 32977* Technical lemma for bnj69 33020. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph 
 <->  ( f `  (/) )  = 
 pred ( X ,  A ,  R )
 )   &    |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph  /\  ps ) )   &    |-  ( th  <->  ( R  FrSe  A 
 /\  X  e.  A  /\  y  e.  trCl ( X ,  A ,  R )  /\  z  e. 
 pred ( y ,  A ,  R ) ) )   &    |-  ( ta  <->  ( m  e. 
 om  /\  n  =  suc  m  /\  p  = 
 suc  n ) )   &    |-  ( et  <->  ( i  e.  n  /\  y  e.  ( f `  i
 ) ) )   &    |-  ( ph'  <->  [. p  /  n ]. ph )   &    |-  ( ps'  <->  [. p  /  n ].
 ps )   &    |-  ( ch'  <->  [. p  /  n ].
 ch )   &    |-  ( ph"  <->  [. G  /  f ]. ph' )   &    |-  ( ps"  <->  [. G  /  f ]. ps' )   &    |-  ( ch"  <->  [. G  /  f ]. ch' )   &    |-  D  =  ( om  \  { (/) } )   &    |-  B  =  { f  |  E. n  e.  D  (
 f  Fn  n  /\  ph 
 /\  ps ) }   &    |-  C  =  U_ y  e.  (
 f `  m )  pred ( y ,  A ,  R )   &    |-  G  =  ( f  u.  { <. n ,  C >. } )   =>    |-  (
 ( R  FrSe  A  /\  X  e.  A )  -> 
 TrFo (  trCl ( X ,  A ,  R ) ,  A ,  R ) )
 
Theorembnj1029 32978 Property of  trCl. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  (
 ( R  FrSe  A  /\  X  e.  A )  -> 
 TrFo (  trCl ( X ,  A ,  R ) ,  A ,  R ) )
 
Theorembnj1033 32979* Technical lemma for bnj69 33020. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph 
 <->  ( f `  (/) )  = 
 pred ( X ,  A ,  R )
 )   &    |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph  /\  ps ) )   &    |-  ( th  <->  ( R  FrSe  A 
 /\  X  e.  A ) )   &    |-  ( ta  <->  ( B  e.  _V 
 /\  TrFo ( B ,  A ,  R )  /\  pred ( X ,  A ,  R )  C_  B ) )   &    |-  ( et 
 <->  z  e.  trCl ( X ,  A ,  R ) )   &    |-  ( ze 
 <->  ( i  e.  n  /\  z  e.  (
 f `  i )
 ) )   &    |-  D  =  ( om  \  { (/) } )   &    |-  K  =  { f  |  E. n  e.  D  (
 f  Fn  n  /\  ph 
 /\  ps ) }   &    |-  ( E. f E. n E. i ( th  /\  ta 
 /\  ch  /\  ze )  ->  z  e.  B )   =>    |-  ( ( th  /\  ta )  ->  trCl ( X ,  A ,  R )  C_  B )
 
Theorembnj1034 32980* Technical lemma for bnj69 33020. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph 
 <->  ( f `  (/) )  = 
 pred ( X ,  A ,  R )
 )   &    |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph  /\  ps ) )   &    |-  ( th  <->  ( R  FrSe  A 
 /\  X  e.  A ) )   &    |-  ( ta  <->  ( B  e.  _V 
 /\  TrFo ( B ,  A ,  R )  /\  pred ( X ,  A ,  R )  C_  B ) )   &    |-  ( ze 
 <->  ( i  e.  n  /\  z  e.  (
 f `  i )
 ) )   &    |-  D  =  ( om  \  { (/) } )   &    |-  K  =  { f  |  E. n  e.  D  (
 f  Fn  n  /\  ph 
 /\  ps ) }   &    |-  ( E. f E. n E. i ( th  /\  ta 
 /\  ch  /\  ze )  ->  z  e.  B )   =>    |-  ( ( th  /\  ta )  ->  trCl ( X ,  A ,  R )  C_  B )
 
Theorembnj1039 32981 Technical lemma for bnj69 33020. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ps 
 <-> 
 A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  ( ps'  <->  [. j  /  i ]. ps )   =>    |-  ( ps'  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )
 
Theorembnj1040 32982* Technical lemma for bnj69 33020. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph'  <->  [. j  /  i ]. ph )   &    |-  ( ps'  <->  [. j  /  i ]. ps )   &    |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph  /\  ps ) )   &    |-  ( ch'  <->  [. j  /  i ]. ch )   =>    |-  ( ch'  <->  ( n  e.  D  /\  f  Fn  n  /\  ph'  /\  ps' ) )
 
Theorembnj1047 32983 Technical lemma for bnj69 33020. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( rh 
 <-> 
 A. j  e.  n  ( j  _E  i  -> 
 [. j  /  i ]. et ) )   &    |-  ( et'  <->  [. j  /  i ]. et )   =>    |-  ( rh  <->  A. j  e.  n  ( j  _E  i  ->  et' ) )
 
Theorembnj1049 32984 Technical lemma for bnj69 33020. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ze 
 <->  ( i  e.  n  /\  z  e.  (
 f `  i )
 ) )   &    |-  ( et  <->  ( ( th  /\ 
 ta  /\  ch  /\  ze )  ->  z  e.  B ) )   =>    |-  ( A. i  e.  n  et  <->  A. i et )
 
Theorembnj1052 32985* Technical lemma for bnj69 33020. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph 
 <->  ( f `  (/) )  = 
 pred ( X ,  A ,  R )
 )   &    |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph  /\  ps ) )   &    |-  ( th  <->  ( R  FrSe  A 
 /\  X  e.  A ) )   &    |-  ( ta  <->  ( B  e.  _V 
 /\  TrFo ( B ,  A ,  R )  /\  pred ( X ,  A ,  R )  C_  B ) )   &    |-  ( ze 
 <->  ( i  e.  n  /\  z  e.  (
 f `  i )
 ) )   &    |-  D  =  ( om  \  { (/) } )   &    |-  K  =  { f  |  E. n  e.  D  (
 f  Fn  n  /\  ph 
 /\  ps ) }   &    |-  ( et 
 <->  ( ( th  /\  ta 
 /\  ch  /\  ze )  ->  z  e.  B ) )   &    |-  ( rh  <->  A. j  e.  n  ( j  _E  i  -> 
 [. j  /  i ]. et ) )   &    |-  (
 ( th  /\  ta  /\  ch 
 /\  ze )  ->  (  _E  Fr  n  /\  A. i  e.  n  ( rh  ->  et ) ) )   =>    |-  ( ( th  /\  ta )  ->  trCl ( X ,  A ,  R )  C_  B )
 
Theorembnj1053 32986* Technical lemma for bnj69 33020. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph 
 <->  ( f `  (/) )  = 
 pred ( X ,  A ,  R )
 )   &    |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph  /\  ps ) )   &    |-  ( th  <->  ( R  FrSe  A 
 /\  X  e.  A ) )   &    |-  ( ta  <->  ( B  e.  _V 
 /\  TrFo ( B ,  A ,  R )  /\  pred ( X ,  A ,  R )  C_  B ) )   &    |-  ( ze 
 <->  ( i  e.  n  /\  z  e.  (
 f `  i )
 ) )   &    |-  D  =  ( om  \  { (/) } )   &    |-  K  =  { f  |  E. n  e.  D  (
 f  Fn  n  /\  ph 
 /\  ps ) }   &    |-  ( et 
 <->  ( ( th  /\  ta 
 /\  ch  /\  ze )  ->  z  e.  B ) )   &    |-  ( rh  <->  A. j  e.  n  ( j  _E  i  -> 
 [. j  /  i ]. et ) )   &    |-  (
 ( th  /\  ta  /\  ch 
 /\  ze )  ->  A. i  e.  n  ( rh  ->  et ) )   =>    |-  ( ( th  /\ 
 ta )  ->  trCl ( X ,  A ,  R )  C_  B )
 
Theorembnj1071 32987 Technical lemma for bnj69 33020. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  D  =  ( om  \  { (/)
 } )   =>    |-  ( n  e.  D  ->  _E  Fr  n )
 
Theorembnj1083 32988 Technical lemma for bnj69 33020. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ch 
 <->  ( n  e.  D  /\  f  Fn  n  /\  ph  /\  ps )
 )   &    |-  K  =  { f  |  E. n  e.  D  ( f  Fn  n  /\  ph  /\  ps ) }   =>    |-  ( f  e.  K  <->  E. n ch )
 
Theorembnj1090 32989* Technical lemma for bnj69 33020. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( et 
 <->  ( ( f  e.  K  /\  i  e. 
 dom  f )  ->  ( f `  i
 )  C_  B )
 )   &    |-  ( rh  <->  A. j  e.  n  ( j  _E  i  -> 
 [. j  /  i ]. et ) )   &    |-  ( et'  <->  [. j  /  i ]. et )   &    |-  ( si  <->  ( ( j  e.  n  /\  j  _E  i )  ->  et' ) )   &    |-  ( ph0  <->  ( i  e.  n  /\  si  /\  f  e.  K  /\  i  e.  dom  f ) )   &    |-  ( ( th  /\ 
 ta  /\  ch  /\  ze )  ->  A. i E. j
 ( ph0  ->  ( f `  i )  C_  B ) )   =>    |-  ( ( th  /\  ta 
 /\  ch  /\  ze )  ->  A. i  e.  n  ( rh  ->  et )
 )
 
Theorembnj1093 32990* Technical lemma for bnj69 33020. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  E. j
 ( ( ( th  /\ 
 ta  /\  ch )  /\  ph0 )  ->  (
 f `  i )  C_  B )   &    |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph  /\  ps ) )   =>    |-  ( ( th  /\  ta 
 /\  ch  /\  ze )  ->  A. i E. j
 ( ph0  ->  ( f `  i )  C_  B ) )
 
Theorembnj1097 32991 Technical lemma for bnj69 33020. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph 
 <->  ( f `  (/) )  = 
 pred ( X ,  A ,  R )
 )   &    |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph  /\  ps ) )   &    |-  ( ta  <->  ( B  e.  _V 
 /\  TrFo ( B ,  A ,  R )  /\  pred ( X ,  A ,  R )  C_  B ) )   =>    |-  ( ( i  =  (/)  /\  ( ( th  /\  ta  /\  ch )  /\  ph0 ) )  ->  ( f `  i
 )  C_  B )
 
Theorembnj1110 32992* Technical lemma for bnj69 33020. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ch 
 <->  ( n  e.  D  /\  f  Fn  n  /\  ph  /\  ps )
 )   &    |-  D  =  ( om  \  { (/) } )   &    |-  ( si 
 <->  ( ( j  e.  n  /\  j  _E  i )  ->  et' ) )   &    |-  ( ph0  <->  ( i  e.  n  /\  si  /\  f  e.  K  /\  i  e.  dom  f ) )   &    |-  ( et'  <->  ( ( f  e.  K  /\  j  e.  dom  f )  ->  ( f `  j
 )  C_  B )
 )   =>    |- 
 E. j ( ( i  =/=  (/)  /\  (
 ( th  /\  ta  /\  ch )  /\  ph0 ) ) 
 ->  ( j  e.  n  /\  i  =  suc  j  /\  ( f `  j )  C_  B ) )
 
Theorembnj1112 32993* Technical lemma for bnj69 33020. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ps 
 <-> 
 A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   =>    |-  ( ps  <->  A. j ( ( j  e.  om  /\  suc  j  e.  n ) 
 ->  ( f `  suc  j )  =  U_ y  e.  ( f `  j
 )  pred ( y ,  A ,  R ) ) )
 
Theorembnj1118 32994* Technical lemma for bnj69 33020. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ps 
 <-> 
 A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph  /\  ps ) )   &    |-  ( ta  <->  ( B  e.  _V 
 /\  TrFo ( B ,  A ,  R )  /\  pred ( X ,  A ,  R )  C_  B ) )   &    |-  D  =  ( om  \  { (/)
 } )   &    |-  ( si  <->  ( ( j  e.  n  /\  j  _E  i )  ->  et' ) )   &    |-  ( ph0  <->  ( i  e.  n  /\  si  /\  f  e.  K  /\  i  e.  dom  f ) )   &    |-  ( et'  <->  ( ( f  e.  K  /\  j  e.  dom  f )  ->  ( f `  j
 )  C_  B )
 )   =>    |- 
 E. j ( ( i  =/=  (/)  /\  (
 ( th  /\  ta  /\  ch )  /\  ph0 ) ) 
 ->  ( f `  i
 )  C_  B )
 
Theorembnj1121 32995 Technical lemma for bnj69 33020. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( th 
 <->  ( R  FrSe  A  /\  X  e.  A )
 )   &    |-  ( ta  <->  ( B  e.  _V 
 /\  TrFo ( B ,  A ,  R )  /\  pred ( X ,  A ,  R )  C_  B ) )   &    |-  ( ch 
 <->  ( n  e.  D  /\  f  Fn  n  /\  ph  /\  ps )
 )   &    |-  ( ze  <->  ( i  e.  n  /\  z  e.  ( f `  i
 ) ) )   &    |-  ( et 
 <->  ( ( f  e.  K  /\  i  e. 
 dom  f )  ->  ( f `  i
 )  C_  B )
 )   &    |-  ( ( th  /\  ta 
 /\  ch  /\  ze )  ->  A. i  e.  n  et )   &    |-  K  =  {
 f  |  E. n  e.  D  ( f  Fn  n  /\  ph  /\  ps ) }   =>    |-  ( ( th  /\  ta 
 /\  ch  /\  ze )  ->  z  e.  B )
 
Theorembnj1123 32996* Technical lemma for bnj69 33020. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ps 
 <-> 
 A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  K  =  {
 f  |  E. n  e.  D  ( f  Fn  n  /\  ph  /\  ps ) }   &    |-  ( et  <->  ( ( f  e.  K  /\  i  e.  dom  f )  ->  ( f `  i
 )  C_  B )
 )   &    |-  ( et'  <->  [. j  /  i ]. et )   =>    |-  ( et'  <->  ( ( f  e.  K  /\  j  e.  dom  f )  ->  ( f `  j
 )  C_  B )
 )
 
Theorembnj1030 32997* Technical lemma for bnj69 33020. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph 
 <->  ( f `  (/) )  = 
 pred ( X ,  A ,  R )
 )   &    |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph  /\  ps ) )   &    |-  ( th  <->  ( R  FrSe  A 
 /\  X  e.  A ) )   &    |-  ( ta  <->  ( B  e.  _V 
 /\  TrFo ( B ,  A ,  R )  /\  pred ( X ,  A ,  R )  C_  B ) )   &    |-  ( ze 
 <->  ( i  e.  n  /\  z  e.  (
 f `  i )
 ) )   &    |-  D  =  ( om  \  { (/) } )   &    |-  K  =  { f  |  E. n  e.  D  (
 f  Fn  n  /\  ph 
 /\  ps ) }   &    |-  ( et 
 <->  ( ( f  e.  K  /\  i  e. 
 dom  f )  ->  ( f `  i
 )  C_  B )
 )   &    |-  ( rh  <->  A. j  e.  n  ( j  _E  i  -> 
 [. j  /  i ]. et ) )   &    |-  ( ph'  <->  [. j  /  i ]. ph )   &    |-  ( ps'  <->  [. j  /  i ]. ps )   &    |-  ( ch'  <->  [. j  /  i ]. ch )   &    |-  ( th'  <->  [. j  /  i ]. th )   &    |-  ( ta'  <->  [. j  /  i ]. ta )   &    |-  ( ze'  <->  [. j  /  i ]. ze )   &    |-  ( et'  <->  [. j  /  i ]. et )   &    |-  ( si  <->  ( ( j  e.  n  /\  j  _E  i )  ->  et' ) )   &    |-  ( ph0  <->  ( i  e.  n  /\  si  /\  f  e.  K  /\  i  e.  dom  f ) )   =>    |-  ( ( th  /\  ta )  ->  trCl ( X ,  A ,  R )  C_  B )
 
Theorembnj1124 32998 Property of  trCl. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( th 
 <->  ( R  FrSe  A  /\  X  e.  A )
 )   &    |-  ( ta  <->  ( B  e.  _V 
 /\  TrFo ( B ,  A ,  R )  /\  pred ( X ,  A ,  R )  C_  B ) )   =>    |-  ( ( th  /\ 
 ta )  ->  trCl ( X ,  A ,  R )  C_  B )
 
Theorembnj1133 32999* Technical lemma for bnj69 33020. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  D  =  ( om  \  { (/)
 } )   &    |-  ( ch  <->  ( n  e.  D  /\  f  Fn  n  /\  ph  /\  ps ) )   &    |-  ( ta  <->  A. j  e.  n  ( j  _E  i  -> 
 [. j  /  i ]. th ) )   &    |-  (
 ( i  e.  n  /\  ta )  ->  th )   =>    |-  ( ch  ->  A. i  e.  n  th )
 
Theorembnj1128 33000* Technical lemma for bnj69 33020. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
 |-  ( ph 
 <->  ( f `  (/) )  = 
 pred ( X ,  A ,  R )
 )   &    |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i
 )  pred ( y ,  A ,  R ) ) )   &    |-  D  =  ( om  \  { (/) } )   &    |-  B  =  { f  |  E. n  e.  D  (
 f  Fn  n  /\  ph 
 /\  ps ) }   &    |-  ( ch 
 <->  ( n  e.  D  /\  f  Fn  n  /\  ph  /\  ps )
 )   &    |-  ( th  <->  ( ch  ->  ( f `  i ) 
 C_  A ) )   &    |-  ( ta  <->  A. j  e.  n  ( j  _E  i  -> 
 [. j  /  i ]. th ) )   &    |-  ( ph'  <->  [. j  /  i ]. ph )   &    |-  ( ps'  <->  [. j  /  i ]. ps )   &    |-  ( ch'  <->  [. j  /  i ]. ch )   &    |-  ( th'  <->  [. j  /  i ]. th )   =>    |-  ( Y  e.  trCl ( X ,  A ,  R )  ->  Y  e.  A )
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