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Theorem bnj153 32894
Description: Technical lemma for bnj852 32935. 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
bnj153.1  |-  ( ph  <->  ( f `  (/) )  = 
pred ( x ,  A ,  R ) )
bnj153.2  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
bnj153.3  |-  D  =  ( om  \  { (/)
} )
bnj153.4  |-  ( th  <->  ( ( R  FrSe  A  /\  x  e.  A
)  ->  E! f
( f  Fn  n  /\  ph  /\  ps )
) )
bnj153.5  |-  ( ta  <->  A. m  e.  D  ( m  _E  n  ->  [. m  /  n ]. th ) )
Assertion
Ref Expression
bnj153  |-  ( n  =  1o  ->  (
( n  e.  D  /\  ta )  ->  th )
)
Distinct variable groups:    A, f,
i, x, y, n    R, f, i, x, y, n    m, n
Allowed substitution hints:    ph( x, y, f, i, m, n)    ps( x, y, f, i, m, n)    th( x, y, f, i, m, n)    ta( x, y, f, i, m, n)    A( m)    D( x, y, f, i, m, n)    R( m)

Proof of Theorem bnj153
Dummy variable  g is distinct from all other variables.
StepHypRef Expression
1 bnj153.1 . 2  |-  ( ph  <->  ( f `  (/) )  = 
pred ( x ,  A ,  R ) )
2 bnj153.2 . 2  |-  ( ps  <->  A. i  e.  om  ( suc  i  e.  n  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
3 bnj153.3 . 2  |-  D  =  ( om  \  { (/)
} )
4 bnj153.4 . 2  |-  ( th  <->  ( ( R  FrSe  A  /\  x  e.  A
)  ->  E! f
( f  Fn  n  /\  ph  /\  ps )
) )
5 bnj153.5 . 2  |-  ( ta  <->  A. m  e.  D  ( m  _E  n  ->  [. m  /  n ]. th ) )
6 biid 236 . 2  |-  ( ( ( R  FrSe  A  /\  x  e.  A
)  ->  ( f  Fn  n  /\  ph  /\  ps ) )  <->  ( ( R  FrSe  A  /\  x  e.  A )  ->  (
f  Fn  n  /\  ph 
/\  ps ) ) )
7 biid 236 . . . 4  |-  ( [. 1o  /  n ]. ph  <->  [. 1o  /  n ]. ph )
81, 7bnj118 32883 . . 3  |-  ( [. 1o  /  n ]. ph  <->  ( f `  (/) )  =  pred ( x ,  A ,  R ) )
98bicomi 202 . 2  |-  ( ( f `  (/) )  = 
pred ( x ,  A ,  R )  <->  [. 1o  /  n ]. ph )
10 bnj105 32734 . . . 4  |-  1o  e.  _V
112, 10bnj92 32876 . . 3  |-  ( [. 1o  /  n ]. ps  <->  A. i  e.  om  ( suc  i  e.  1o  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
1211bicomi 202 . 2  |-  ( A. i  e.  om  ( suc  i  e.  1o  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) )  <->  [. 1o  /  n ]. ps )
13 biid 236 . 2  |-  ( [. 1o  /  n ]. th  <->  [. 1o  /  n ]. th )
14 biid 236 . 2  |-  ( ( ( R  FrSe  A  /\  x  e.  A
)  ->  E. f
( f  Fn  1o  /\  ( f `  (/) )  = 
pred ( x ,  A ,  R )  /\  A. i  e. 
om  ( suc  i  e.  1o  ->  ( f `  suc  i )  = 
U_ y  e.  ( f `  i ) 
pred ( y ,  A ,  R ) ) ) )  <->  ( ( R  FrSe  A  /\  x  e.  A )  ->  E. f
( f  Fn  1o  /\  ( f `  (/) )  = 
pred ( x ,  A ,  R )  /\  A. i  e. 
om  ( suc  i  e.  1o  ->  ( f `  suc  i )  = 
U_ y  e.  ( f `  i ) 
pred ( y ,  A ,  R ) ) ) ) )
15 biid 236 . 2  |-  ( ( ( R  FrSe  A  /\  x  e.  A
)  ->  E* f
( f  Fn  1o  /\  ( f `  (/) )  = 
pred ( x ,  A ,  R )  /\  A. i  e. 
om  ( suc  i  e.  1o  ->  ( f `  suc  i )  = 
U_ y  e.  ( f `  i ) 
pred ( y ,  A ,  R ) ) ) )  <->  ( ( R  FrSe  A  /\  x  e.  A )  ->  E* f ( f  Fn  1o  /\  ( f `
 (/) )  =  pred ( x ,  A ,  R )  /\  A. i  e.  om  ( suc  i  e.  1o  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) ) ) )
16 biid 236 . . . . 5  |-  ( [. 1o  /  n ]. (
( R  FrSe  A  /\  x  e.  A
)  ->  ( f  Fn  n  /\  ph  /\  ps ) )  <->  [. 1o  /  n ]. ( ( R 
FrSe  A  /\  x  e.  A )  ->  (
f  Fn  n  /\  ph 
/\  ps ) ) )
17 biid 236 . . . . 5  |-  ( [. 1o  /  n ]. ps  <->  [. 1o  /  n ]. ps )
186, 16, 7, 17bnj121 32884 . . . 4  |-  ( [. 1o  /  n ]. (
( R  FrSe  A  /\  x  e.  A
)  ->  ( f  Fn  n  /\  ph  /\  ps ) )  <->  ( ( R  FrSe  A  /\  x  e.  A )  ->  (
f  Fn  1o  /\  [. 1o  /  n ]. ph 
/\  [. 1o  /  n ]. ps ) ) )
198anbi2i 694 . . . . . . 7  |-  ( ( f  Fn  1o  /\  [. 1o  /  n ]. ph )  <->  ( f  Fn  1o  /\  ( f `
 (/) )  =  pred ( x ,  A ,  R ) ) )
2019, 11anbi12i 697 . . . . . 6  |-  ( ( ( f  Fn  1o  /\ 
[. 1o  /  n ]. ph )  /\  [. 1o  /  n ]. ps )  <->  ( ( f  Fn  1o  /\  ( f `  (/) )  = 
pred ( x ,  A ,  R ) )  /\  A. i  e.  om  ( suc  i  e.  1o  ->  ( f `  suc  i )  = 
U_ y  e.  ( f `  i ) 
pred ( y ,  A ,  R ) ) ) )
21 df-3an 970 . . . . . 6  |-  ( ( f  Fn  1o  /\  [. 1o  /  n ]. ph 
/\  [. 1o  /  n ]. ps )  <->  ( (
f  Fn  1o  /\  [. 1o  /  n ]. ph )  /\  [. 1o  /  n ]. ps )
)
22 df-3an 970 . . . . . 6  |-  ( ( f  Fn  1o  /\  ( f `  (/) )  = 
pred ( x ,  A ,  R )  /\  A. i  e. 
om  ( suc  i  e.  1o  ->  ( f `  suc  i )  = 
U_ y  e.  ( f `  i ) 
pred ( y ,  A ,  R ) ) )  <->  ( (
f  Fn  1o  /\  ( f `  (/) )  = 
pred ( x ,  A ,  R ) )  /\  A. i  e.  om  ( suc  i  e.  1o  ->  ( f `  suc  i )  = 
U_ y  e.  ( f `  i ) 
pred ( y ,  A ,  R ) ) ) )
2320, 21, 223bitr4i 277 . . . . 5  |-  ( ( f  Fn  1o  /\  [. 1o  /  n ]. ph 
/\  [. 1o  /  n ]. ps )  <->  ( f  Fn  1o  /\  ( f `
 (/) )  =  pred ( x ,  A ,  R )  /\  A. i  e.  om  ( suc  i  e.  1o  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) ) )
2423imbi2i 312 . . . 4  |-  ( ( ( R  FrSe  A  /\  x  e.  A
)  ->  ( f  Fn  1o  /\  [. 1o  /  n ]. ph  /\  [. 1o  /  n ]. ps ) )  <->  ( ( R  FrSe  A  /\  x  e.  A )  ->  (
f  Fn  1o  /\  ( f `  (/) )  = 
pred ( x ,  A ,  R )  /\  A. i  e. 
om  ( suc  i  e.  1o  ->  ( f `  suc  i )  = 
U_ y  e.  ( f `  i ) 
pred ( y ,  A ,  R ) ) ) ) )
2518, 24bitri 249 . . 3  |-  ( [. 1o  /  n ]. (
( R  FrSe  A  /\  x  e.  A
)  ->  ( f  Fn  n  /\  ph  /\  ps ) )  <->  ( ( R  FrSe  A  /\  x  e.  A )  ->  (
f  Fn  1o  /\  ( f `  (/) )  = 
pred ( x ,  A ,  R )  /\  A. i  e. 
om  ( suc  i  e.  1o  ->  ( f `  suc  i )  = 
U_ y  e.  ( f `  i ) 
pred ( y ,  A ,  R ) ) ) ) )
2625bicomi 202 . 2  |-  ( ( ( R  FrSe  A  /\  x  e.  A
)  ->  ( f  Fn  1o  /\  ( f `
 (/) )  =  pred ( x ,  A ,  R )  /\  A. i  e.  om  ( suc  i  e.  1o  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) ) )  <->  [. 1o  /  n ]. ( ( R  FrSe  A  /\  x  e.  A
)  ->  ( f  Fn  n  /\  ph  /\  ps ) ) )
27 eqid 2462 . 2  |-  { <. (/)
,  pred ( x ,  A ,  R )
>. }  =  { <. (/)
,  pred ( x ,  A ,  R )
>. }
28 biid 236 . 2  |-  ( [. { <. (/) ,  pred (
x ,  A ,  R ) >. }  / 
f ]. ( f `  (/) )  =  pred (
x ,  A ,  R )  <->  [. { <. (/)
,  pred ( x ,  A ,  R )
>. }  /  f ]. ( f `  (/) )  = 
pred ( x ,  A ,  R ) )
29 biid 236 . 2  |-  ( [. { <. (/) ,  pred (
x ,  A ,  R ) >. }  / 
f ]. A. i  e. 
om  ( suc  i  e.  1o  ->  ( f `  suc  i )  = 
U_ y  e.  ( f `  i ) 
pred ( y ,  A ,  R ) )  <->  [. { <. (/) ,  pred ( x ,  A ,  R ) >. }  / 
f ]. A. i  e. 
om  ( suc  i  e.  1o  ->  ( f `  suc  i )  = 
U_ y  e.  ( f `  i ) 
pred ( y ,  A ,  R ) ) )
3026sbcbii 3386 . . 3  |-  ( [. { <. (/) ,  pred (
x ,  A ,  R ) >. }  / 
f ]. ( ( R 
FrSe  A  /\  x  e.  A )  ->  (
f  Fn  1o  /\  ( f `  (/) )  = 
pred ( x ,  A ,  R )  /\  A. i  e. 
om  ( suc  i  e.  1o  ->  ( f `  suc  i )  = 
U_ y  e.  ( f `  i ) 
pred ( y ,  A ,  R ) ) ) )  <->  [. { <. (/)
,  pred ( x ,  A ,  R )
>. }  /  f ]. [. 1o  /  n ]. ( ( R  FrSe  A  /\  x  e.  A
)  ->  ( f  Fn  n  /\  ph  /\  ps ) ) )
31 biid 236 . . . . 5  |-  ( [. { <. (/) ,  pred (
x ,  A ,  R ) >. }  / 
f ]. [. 1o  /  n ]. ph  <->  [. { <. (/)
,  pred ( x ,  A ,  R )
>. }  /  f ]. [. 1o  /  n ]. ph )
32 biid 236 . . . . 5  |-  ( [. { <. (/) ,  pred (
x ,  A ,  R ) >. }  / 
f ]. [. 1o  /  n ]. ps  <->  [. { <. (/)
,  pred ( x ,  A ,  R )
>. }  /  f ]. [. 1o  /  n ]. ps )
33 biid 236 . . . . 5  |-  ( [. { <. (/) ,  pred (
x ,  A ,  R ) >. }  / 
f ]. [. 1o  /  n ]. ( ( R 
FrSe  A  /\  x  e.  A )  ->  (
f  Fn  n  /\  ph 
/\  ps ) )  <->  [. { <. (/)
,  pred ( x ,  A ,  R )
>. }  /  f ]. [. 1o  /  n ]. ( ( R  FrSe  A  /\  x  e.  A
)  ->  ( f  Fn  n  /\  ph  /\  ps ) ) )
3427, 31, 32, 33, 18bnj124 32885 . . . 4  |-  ( [. { <. (/) ,  pred (
x ,  A ,  R ) >. }  / 
f ]. [. 1o  /  n ]. ( ( R 
FrSe  A  /\  x  e.  A )  ->  (
f  Fn  n  /\  ph 
/\  ps ) )  <->  ( ( R  FrSe  A  /\  x  e.  A )  ->  ( { <. (/) ,  pred (
x ,  A ,  R ) >. }  Fn  1o  /\  [. { <. (/)
,  pred ( x ,  A ,  R )
>. }  /  f ]. [. 1o  /  n ]. ph 
/\  [. { <. (/) ,  pred ( x ,  A ,  R ) >. }  / 
f ]. [. 1o  /  n ]. ps ) ) )
351, 7, 31, 27bnj125 32886 . . . . . . . 8  |-  ( [. { <. (/) ,  pred (
x ,  A ,  R ) >. }  / 
f ]. [. 1o  /  n ]. ph  <->  ( { <.
(/) ,  pred ( x ,  A ,  R
) >. } `  (/) )  = 
pred ( x ,  A ,  R ) )
3635anbi2i 694 . . . . . . 7  |-  ( ( { <. (/) ,  pred (
x ,  A ,  R ) >. }  Fn  1o  /\  [. { <. (/)
,  pred ( x ,  A ,  R )
>. }  /  f ]. [. 1o  /  n ]. ph )  <->  ( { <. (/)
,  pred ( x ,  A ,  R )
>. }  Fn  1o  /\  ( { <. (/) ,  pred (
x ,  A ,  R ) >. } `  (/) )  =  pred (
x ,  A ,  R ) ) )
372, 17, 32, 27bnj126 32887 . . . . . . 7  |-  ( [. { <. (/) ,  pred (
x ,  A ,  R ) >. }  / 
f ]. [. 1o  /  n ]. ps  <->  A. i  e.  om  ( suc  i  e.  1o  ->  ( { <.
(/) ,  pred ( x ,  A ,  R
) >. } `  suc  i )  =  U_ y  e.  ( { <.
(/) ,  pred ( x ,  A ,  R
) >. } `  i
)  pred ( y ,  A ,  R ) ) )
3836, 37anbi12i 697 . . . . . 6  |-  ( ( ( { <. (/) ,  pred ( x ,  A ,  R ) >. }  Fn  1o  /\  [. { <. (/)
,  pred ( x ,  A ,  R )
>. }  /  f ]. [. 1o  /  n ]. ph )  /\  [. { <.
(/) ,  pred ( x ,  A ,  R
) >. }  /  f ]. [. 1o  /  n ]. ps )  <->  ( ( { <. (/) ,  pred (
x ,  A ,  R ) >. }  Fn  1o  /\  ( { <. (/)
,  pred ( x ,  A ,  R )
>. } `  (/) )  = 
pred ( x ,  A ,  R ) )  /\  A. i  e.  om  ( suc  i  e.  1o  ->  ( { <.
(/) ,  pred ( x ,  A ,  R
) >. } `  suc  i )  =  U_ y  e.  ( { <.
(/) ,  pred ( x ,  A ,  R
) >. } `  i
)  pred ( y ,  A ,  R ) ) ) )
39 df-3an 970 . . . . . 6  |-  ( ( { <. (/) ,  pred (
x ,  A ,  R ) >. }  Fn  1o  /\  [. { <. (/)
,  pred ( x ,  A ,  R )
>. }  /  f ]. [. 1o  /  n ]. ph 
/\  [. { <. (/) ,  pred ( x ,  A ,  R ) >. }  / 
f ]. [. 1o  /  n ]. ps )  <->  ( ( { <. (/) ,  pred (
x ,  A ,  R ) >. }  Fn  1o  /\  [. { <. (/)
,  pred ( x ,  A ,  R )
>. }  /  f ]. [. 1o  /  n ]. ph )  /\  [. { <.
(/) ,  pred ( x ,  A ,  R
) >. }  /  f ]. [. 1o  /  n ]. ps ) )
40 df-3an 970 . . . . . 6  |-  ( ( { <. (/) ,  pred (
x ,  A ,  R ) >. }  Fn  1o  /\  ( { <. (/)
,  pred ( x ,  A ,  R )
>. } `  (/) )  = 
pred ( x ,  A ,  R )  /\  A. i  e. 
om  ( suc  i  e.  1o  ->  ( { <.
(/) ,  pred ( x ,  A ,  R
) >. } `  suc  i )  =  U_ y  e.  ( { <.
(/) ,  pred ( x ,  A ,  R
) >. } `  i
)  pred ( y ,  A ,  R ) ) )  <->  ( ( { <. (/) ,  pred (
x ,  A ,  R ) >. }  Fn  1o  /\  ( { <. (/)
,  pred ( x ,  A ,  R )
>. } `  (/) )  = 
pred ( x ,  A ,  R ) )  /\  A. i  e.  om  ( suc  i  e.  1o  ->  ( { <.
(/) ,  pred ( x ,  A ,  R
) >. } `  suc  i )  =  U_ y  e.  ( { <.
(/) ,  pred ( x ,  A ,  R
) >. } `  i
)  pred ( y ,  A ,  R ) ) ) )
4138, 39, 403bitr4i 277 . . . . 5  |-  ( ( { <. (/) ,  pred (
x ,  A ,  R ) >. }  Fn  1o  /\  [. { <. (/)
,  pred ( x ,  A ,  R )
>. }  /  f ]. [. 1o  /  n ]. ph 
/\  [. { <. (/) ,  pred ( x ,  A ,  R ) >. }  / 
f ]. [. 1o  /  n ]. ps )  <->  ( { <.
(/) ,  pred ( x ,  A ,  R
) >. }  Fn  1o  /\  ( { <. (/) ,  pred ( x ,  A ,  R ) >. } `  (/) )  =  pred (
x ,  A ,  R )  /\  A. i  e.  om  ( suc  i  e.  1o  ->  ( { <. (/) ,  pred ( x ,  A ,  R ) >. } `  suc  i )  =  U_ y  e.  ( { <.
(/) ,  pred ( x ,  A ,  R
) >. } `  i
)  pred ( y ,  A ,  R ) ) ) )
4241imbi2i 312 . . . 4  |-  ( ( ( R  FrSe  A  /\  x  e.  A
)  ->  ( { <.
(/) ,  pred ( x ,  A ,  R
) >. }  Fn  1o  /\ 
[. { <. (/) ,  pred ( x ,  A ,  R ) >. }  / 
f ]. [. 1o  /  n ]. ph  /\  [. { <.
(/) ,  pred ( x ,  A ,  R
) >. }  /  f ]. [. 1o  /  n ]. ps ) )  <->  ( ( R  FrSe  A  /\  x  e.  A )  ->  ( { <. (/) ,  pred (
x ,  A ,  R ) >. }  Fn  1o  /\  ( { <. (/)
,  pred ( x ,  A ,  R )
>. } `  (/) )  = 
pred ( x ,  A ,  R )  /\  A. i  e. 
om  ( suc  i  e.  1o  ->  ( { <.
(/) ,  pred ( x ,  A ,  R
) >. } `  suc  i )  =  U_ y  e.  ( { <.
(/) ,  pred ( x ,  A ,  R
) >. } `  i
)  pred ( y ,  A ,  R ) ) ) ) )
4334, 42bitri 249 . . 3  |-  ( [. { <. (/) ,  pred (
x ,  A ,  R ) >. }  / 
f ]. [. 1o  /  n ]. ( ( R 
FrSe  A  /\  x  e.  A )  ->  (
f  Fn  n  /\  ph 
/\  ps ) )  <->  ( ( R  FrSe  A  /\  x  e.  A )  ->  ( { <. (/) ,  pred (
x ,  A ,  R ) >. }  Fn  1o  /\  ( { <. (/)
,  pred ( x ,  A ,  R )
>. } `  (/) )  = 
pred ( x ,  A ,  R )  /\  A. i  e. 
om  ( suc  i  e.  1o  ->  ( { <.
(/) ,  pred ( x ,  A ,  R
) >. } `  suc  i )  =  U_ y  e.  ( { <.
(/) ,  pred ( x ,  A ,  R
) >. } `  i
)  pred ( y ,  A ,  R ) ) ) ) )
4430, 43bitr2i 250 . 2  |-  ( ( ( R  FrSe  A  /\  x  e.  A
)  ->  ( { <.
(/) ,  pred ( x ,  A ,  R
) >. }  Fn  1o  /\  ( { <. (/) ,  pred ( x ,  A ,  R ) >. } `  (/) )  =  pred (
x ,  A ,  R )  /\  A. i  e.  om  ( suc  i  e.  1o  ->  ( { <. (/) ,  pred ( x ,  A ,  R ) >. } `  suc  i )  =  U_ y  e.  ( { <.
(/) ,  pred ( x ,  A ,  R
) >. } `  i
)  pred ( y ,  A ,  R ) ) ) )  <->  [. { <. (/)
,  pred ( x ,  A ,  R )
>. }  /  f ]. ( ( R  FrSe  A  /\  x  e.  A
)  ->  ( f  Fn  1o  /\  ( f `
 (/) )  =  pred ( x ,  A ,  R )  /\  A. i  e.  om  ( suc  i  e.  1o  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) ) ) )
45 biid 236 . 2  |-  ( ( f  Fn  1o  /\  ( f `  (/) )  = 
pred ( x ,  A ,  R )  /\  A. i  e. 
om  ( suc  i  e.  1o  ->  ( f `  suc  i )  = 
U_ y  e.  ( f `  i ) 
pred ( y ,  A ,  R ) ) )  <->  ( f  Fn  1o  /\  ( f `
 (/) )  =  pred ( x ,  A ,  R )  /\  A. i  e.  om  ( suc  i  e.  1o  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) ) )
46 biid 236 . . . . 5  |-  ( ( f  Fn  1o  /\  [. 1o  /  n ]. ph 
/\  [. 1o  /  n ]. ps )  <->  ( f  Fn  1o  /\  [. 1o  /  n ]. ph  /\  [. 1o  /  n ]. ps ) )
47 biid 236 . . . . 5  |-  ( [. g  /  f ]. (
f  Fn  1o  /\  [. 1o  /  n ]. ph 
/\  [. 1o  /  n ]. ps )  <->  [. g  / 
f ]. ( f  Fn  1o  /\  [. 1o  /  n ]. ph  /\  [. 1o  /  n ]. ps ) )
48 biid 236 . . . . 5  |-  ( [. g  /  f ]. [. 1o  /  n ]. ph  <->  [. g  / 
f ]. [. 1o  /  n ]. ph )
49 biid 236 . . . . 5  |-  ( [. g  /  f ]. [. 1o  /  n ]. ps  <->  [. g  / 
f ]. [. 1o  /  n ]. ps )
5046, 47, 48, 49bnj156 32740 . . . 4  |-  ( [. g  /  f ]. (
f  Fn  1o  /\  [. 1o  /  n ]. ph 
/\  [. 1o  /  n ]. ps )  <->  ( g  Fn  1o  /\  [. g  /  f ]. [. 1o  /  n ]. ph  /\  [. g  /  f ]. [. 1o  /  n ]. ps ) )
5148, 8bnj154 32892 . . . . . . 7  |-  ( [. g  /  f ]. [. 1o  /  n ]. ph  <->  ( g `  (/) )  =  pred ( x ,  A ,  R ) )
5251anbi2i 694 . . . . . 6  |-  ( ( g  Fn  1o  /\  [. g  /  f ]. [. 1o  /  n ]. ph )  <->  ( g  Fn  1o  /\  ( g `
 (/) )  =  pred ( x ,  A ,  R ) ) )
5317, 11bitri 249 . . . . . . 7  |-  ( [. 1o  /  n ]. ps  <->  A. i  e.  om  ( suc  i  e.  1o  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
5449, 53bnj155 32893 . . . . . 6  |-  ( [. g  /  f ]. [. 1o  /  n ]. ps  <->  A. i  e.  om  ( suc  i  e.  1o  ->  ( g `  suc  i )  = 
U_ y  e.  ( g `  i ) 
pred ( y ,  A ,  R ) ) )
5552, 54anbi12i 697 . . . . 5  |-  ( ( ( g  Fn  1o  /\ 
[. g  /  f ]. [. 1o  /  n ]. ph )  /\  [. g  /  f ]. [. 1o  /  n ]. ps )  <->  ( ( g  Fn  1o  /\  ( g `  (/) )  = 
pred ( x ,  A ,  R ) )  /\  A. i  e.  om  ( suc  i  e.  1o  ->  ( g `  suc  i )  = 
U_ y  e.  ( g `  i ) 
pred ( y ,  A ,  R ) ) ) )
56 df-3an 970 . . . . 5  |-  ( ( g  Fn  1o  /\  [. g  /  f ]. [. 1o  /  n ]. ph 
/\  [. g  /  f ]. [. 1o  /  n ]. ps )  <->  ( (
g  Fn  1o  /\  [. g  /  f ]. [. 1o  /  n ]. ph )  /\  [. g  /  f ]. [. 1o  /  n ]. ps )
)
57 df-3an 970 . . . . 5  |-  ( ( g  Fn  1o  /\  ( g `  (/) )  = 
pred ( x ,  A ,  R )  /\  A. i  e. 
om  ( suc  i  e.  1o  ->  ( g `  suc  i )  = 
U_ y  e.  ( g `  i ) 
pred ( y ,  A ,  R ) ) )  <->  ( (
g  Fn  1o  /\  ( g `  (/) )  = 
pred ( x ,  A ,  R ) )  /\  A. i  e.  om  ( suc  i  e.  1o  ->  ( g `  suc  i )  = 
U_ y  e.  ( g `  i ) 
pred ( y ,  A ,  R ) ) ) )
5855, 56, 573bitr4i 277 . . . 4  |-  ( ( g  Fn  1o  /\  [. g  /  f ]. [. 1o  /  n ]. ph 
/\  [. g  /  f ]. [. 1o  /  n ]. ps )  <->  ( g  Fn  1o  /\  ( g `
 (/) )  =  pred ( x ,  A ,  R )  /\  A. i  e.  om  ( suc  i  e.  1o  ->  ( g `  suc  i )  =  U_ y  e.  ( g `  i )  pred (
y ,  A ,  R ) ) ) )
5950, 58bitri 249 . . 3  |-  ( [. g  /  f ]. (
f  Fn  1o  /\  [. 1o  /  n ]. ph 
/\  [. 1o  /  n ]. ps )  <->  ( g  Fn  1o  /\  ( g `
 (/) )  =  pred ( x ,  A ,  R )  /\  A. i  e.  om  ( suc  i  e.  1o  ->  ( g `  suc  i )  =  U_ y  e.  ( g `  i )  pred (
y ,  A ,  R ) ) ) )
6023sbcbii 3386 . . 3  |-  ( [. g  /  f ]. (
f  Fn  1o  /\  [. 1o  /  n ]. ph 
/\  [. 1o  /  n ]. ps )  <->  [. g  / 
f ]. ( f  Fn  1o  /\  ( f `
 (/) )  =  pred ( x ,  A ,  R )  /\  A. i  e.  om  ( suc  i  e.  1o  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) ) )
6159, 60bitr3i 251 . 2  |-  ( ( g  Fn  1o  /\  ( g `  (/) )  = 
pred ( x ,  A ,  R )  /\  A. i  e. 
om  ( suc  i  e.  1o  ->  ( g `  suc  i )  = 
U_ y  e.  ( g `  i ) 
pred ( y ,  A ,  R ) ) )  <->  [. g  / 
f ]. ( f  Fn  1o  /\  ( f `
 (/) )  =  pred ( x ,  A ,  R )  /\  A. i  e.  om  ( suc  i  e.  1o  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) ) )
62 biid 236 . 2  |-  ( [. g  /  f ]. (
f `  (/) )  = 
pred ( x ,  A ,  R )  <->  [. g  /  f ]. ( f `  (/) )  = 
pred ( x ,  A ,  R ) )
63 biid 236 . 2  |-  ( [. g  /  f ]. A. i  e.  om  ( suc  i  e.  1o  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) )  <->  [. g  / 
f ]. A. i  e. 
om  ( suc  i  e.  1o  ->  ( f `  suc  i )  = 
U_ y  e.  ( f `  i ) 
pred ( y ,  A ,  R ) ) )
641, 2, 3, 4, 5, 6, 9, 12, 13, 14, 15, 26, 27, 28, 29, 44, 45, 61, 62, 63bnj151 32891 1  |-  ( n  =  1o  ->  (
( n  e.  D  /\  ta )  ->  th )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 968    = wceq 1374   E.wex 1591    e. wcel 1762   E!weu 2270   E*wmo 2271   A.wral 2809   [.wsbc 3326    \ cdif 3468   (/)c0 3780   {csn 4022   <.cop 4028   U_ciun 4320   class class class wbr 4442    _E cep 4784   suc csuc 4875    Fn wfn 5576   ` cfv 5581   omcom 6673   1oc1o 7115    predc-bnj14 32697    FrSe w-bnj15 32701
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-reu 2816  df-rab 2818  df-v 3110  df-sbc 3327  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-1o 7122  df-bnj13 32700  df-bnj15 32702
This theorem is referenced by:  bnj852  32935
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