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Theorem bnj938 29757
Description: Technical lemma for bnj69 29828. 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.)
Hypotheses
Ref Expression
bnj938.1  |-  D  =  ( om  \  { (/)
} )
bnj938.2  |-  ( ta  <->  ( f  Fn  m  /\  ph' 
/\  ps' ) )
bnj938.3  |-  ( si  <->  ( m  e.  D  /\  n  =  suc  m  /\  p  e.  m )
)
bnj938.4  |-  ( ph'  <->  (
f `  (/) )  = 
pred ( X ,  A ,  R )
)
bnj938.5  |-  ( ps'  <->  A. i  e.  om  ( suc  i  e.  m  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
Assertion
Ref Expression
bnj938  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  ta  /\  si )  ->  U_ y  e.  (
f `  p )  pred ( y ,  A ,  R )  e.  _V )
Distinct variable groups:    A, i, p, y    R, i, p, y    f, i, p, y    i, m, p
Allowed substitution hints:    ta( y, f, i, m, n, p)    si( y, f, i, m, n, p)    A( f, m, n)    D( y, f, i, m, n, p)    R( f, m, n)    X( y, f, i, m, n, p)    ph'( y, f, i, m, n, p)    ps'( y, f, i, m, n, p)

Proof of Theorem bnj938
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 elisset 3091 . . 3  |-  ( X  e.  A  ->  E. x  x  =  X )
21bnj706 29573 . 2  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  ta  /\  si )  ->  E. x  x  =  X )
3 bnj291 29525 . . . . . 6  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  ta  /\  si )  <->  ( ( R  FrSe  A  /\  ta  /\ 
si )  /\  X  e.  A ) )
43simplbi 461 . . . . 5  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  ta  /\  si )  -> 
( R  FrSe  A  /\  ta  /\  si )
)
5 bnj602 29735 . . . . . . . . . 10  |-  ( x  =  X  ->  pred (
x ,  A ,  R )  =  pred ( X ,  A ,  R ) )
65eqeq2d 2436 . . . . . . . . 9  |-  ( x  =  X  ->  (
( f `  (/) )  = 
pred ( x ,  A ,  R )  <-> 
( f `  (/) )  = 
pred ( X ,  A ,  R )
) )
7 bnj938.4 . . . . . . . . 9  |-  ( ph'  <->  (
f `  (/) )  = 
pred ( X ,  A ,  R )
)
86, 7syl6bbr 266 . . . . . . . 8  |-  ( x  =  X  ->  (
( f `  (/) )  = 
pred ( x ,  A ,  R )  <->  ph' ) )
983anbi2d 1340 . . . . . . 7  |-  ( x  =  X  ->  (
( f  Fn  m  /\  ( f `  (/) )  = 
pred ( x ,  A ,  R )  /\  ps' )  <->  ( f  Fn  m  /\  ph'  /\  ps' ) ) )
10 bnj938.2 . . . . . . 7  |-  ( ta  <->  ( f  Fn  m  /\  ph' 
/\  ps' ) )
119, 10syl6bbr 266 . . . . . 6  |-  ( x  =  X  ->  (
( f  Fn  m  /\  ( f `  (/) )  = 
pred ( x ,  A ,  R )  /\  ps' )  <->  ta )
)
12113anbi2d 1340 . . . . 5  |-  ( x  =  X  ->  (
( R  FrSe  A  /\  ( f  Fn  m  /\  ( f `  (/) )  = 
pred ( x ,  A ,  R )  /\  ps' )  /\  si ) 
<->  ( R  FrSe  A  /\  ta  /\  si )
) )
134, 12syl5ibr 224 . . . 4  |-  ( x  =  X  ->  (
( R  FrSe  A  /\  X  e.  A  /\  ta  /\  si )  ->  ( R  FrSe  A  /\  ( f  Fn  m  /\  ( f `  (/) )  = 
pred ( x ,  A ,  R )  /\  ps' )  /\  si ) ) )
14 bnj938.1 . . . . 5  |-  D  =  ( om  \  { (/)
} )
15 biid 239 . . . . 5  |-  ( ( f  Fn  m  /\  ( f `  (/) )  = 
pred ( x ,  A ,  R )  /\  ps' )  <->  ( f  Fn  m  /\  (
f `  (/) )  = 
pred ( x ,  A ,  R )  /\  ps' ) )
16 bnj938.3 . . . . 5  |-  ( si  <->  ( m  e.  D  /\  n  =  suc  m  /\  p  e.  m )
)
17 biid 239 . . . . 5  |-  ( ( f `  (/) )  = 
pred ( x ,  A ,  R )  <-> 
( f `  (/) )  = 
pred ( x ,  A ,  R ) )
18 bnj938.5 . . . . 5  |-  ( ps'  <->  A. i  e.  om  ( suc  i  e.  m  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
1914, 15, 16, 17, 18bnj546 29716 . . . 4  |-  ( ( R  FrSe  A  /\  ( f  Fn  m  /\  ( f `  (/) )  = 
pred ( x ,  A ,  R )  /\  ps' )  /\  si )  ->  U_ y  e.  ( f `  p ) 
pred ( y ,  A ,  R )  e.  _V )
2013, 19syl6 34 . . 3  |-  ( x  =  X  ->  (
( R  FrSe  A  /\  X  e.  A  /\  ta  /\  si )  ->  U_ y  e.  ( f `  p ) 
pred ( y ,  A ,  R )  e.  _V ) )
2120exlimiv 1770 . 2  |-  ( E. x  x  =  X  ->  ( ( R 
FrSe  A  /\  X  e.  A  /\  ta  /\  si )  ->  U_ y  e.  ( f `  p
)  pred ( y ,  A ,  R )  e.  _V ) )
222, 21mpcom 37 1  |-  ( ( R  FrSe  A  /\  X  e.  A  /\  ta  /\  si )  ->  U_ y  e.  (
f `  p )  pred ( y ,  A ,  R )  e.  _V )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ w3a 982    = wceq 1437   E.wex 1657    e. wcel 1872   A.wral 2771   _Vcvv 3080    \ cdif 3433   (/)c0 3761   {csn 3998   U_ciun 4299   suc csuc 5444    Fn wfn 5596   ` cfv 5601   omcom 6707    /\ w-bnj17 29500    predc-bnj14 29502    FrSe w-bnj15 29506
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-rep 4536  ax-sep 4546  ax-nul 4555  ax-pr 4660  ax-un 6598
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-reu 2778  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-om 6708  df-bnj17 29501  df-bnj14 29503  df-bnj13 29505  df-bnj15 29507
This theorem is referenced by:  bnj944  29758  bnj969  29766
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