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Theorem bnj153 31871
Description: Technical lemma for bnj852 31912. 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 31860 . . 3  |-  ( [. 1o  /  n ]. ph  <->  ( f `  (/) )  =  pred ( x ,  A ,  R ) )
98bicomi 202 . 2  |-  ( ( f `  (/) )  = 
pred ( x ,  A ,  R )  <->  [. 1o  /  n ]. ph )
10 bnj105 31711 . . . 4  |-  1o  e.  _V
112, 10bnj92 31853 . . 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 31861 . . . 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 967 . . . . . 6  |-  ( ( f  Fn  1o  /\  [. 1o  /  n ]. ph 
/\  [. 1o  /  n ]. ps )  <->  ( (
f  Fn  1o  /\  [. 1o  /  n ]. ph )  /\  [. 1o  /  n ]. ps )
)
22 df-3an 967 . . . . . 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 2442 . 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 3245 . . 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 31862 . . . 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 31863 . . . . . . . 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 31864 . . . . . . 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 967 . . . . . 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 967 . . . . . 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 31717 . . . 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 31869 . . . . . . 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 31870 . . . . . 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 967 . . . . 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 967 . . . . 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 3245 . . 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 31868 1  |-  ( n  =  1o  ->  (
( n  e.  D  /\  ta )  ->  th )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369   E.wex 1586    e. wcel 1756   E!weu 2253   E*wmo 2254   A.wral 2714   [.wsbc 3185    \ cdif 3324   (/)c0 3636   {csn 3876   <.cop 3882   U_ciun 4170   class class class wbr 4291    _E cep 4629   suc csuc 4720    Fn wfn 5412   ` cfv 5417   omcom 6475   1oc1o 6912    predc-bnj14 31674    FrSe w-bnj15 31678
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-reu 2721  df-rab 2723  df-v 2973  df-sbc 3186  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-id 4635  df-suc 4724  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-1o 6919  df-bnj13 31677  df-bnj15 31679
This theorem is referenced by:  bnj852  31912
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