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Theorem bnj153 28912
Description: Technical lemma for bnj852 28953. 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 227 . 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 227 . . . 4  |-  ( [. 1o  /  n ]. ph  <->  [. 1o  /  n ]. ph )
81, 7bnj118 28901 . . 3  |-  ( [. 1o  /  n ]. ph  <->  ( f `  (/) )  =  pred ( x ,  A ,  R ) )
98bicomi 193 . 2  |-  ( ( f `  (/) )  = 
pred ( x ,  A ,  R )  <->  [. 1o  /  n ]. ph )
10 biid 227 . . . 4  |-  ( [. 1o  /  n ]. ps  <->  [. 1o  /  n ]. ps )
11 bnj105 28750 . . . . 5  |-  1o  e.  _V
122, 11bnj92 28894 . . . 4  |-  ( [. 1o  /  n ]. ps  <->  A. i  e.  om  ( suc  i  e.  1o  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
1310, 12bitri 240 . . 3  |-  ( [. 1o  /  n ]. ps  <->  A. i  e.  om  ( suc  i  e.  1o  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) ) )
1413bicomi 193 . 2  |-  ( A. i  e.  om  ( suc  i  e.  1o  ->  ( f `  suc  i )  =  U_ y  e.  ( f `  i )  pred (
y ,  A ,  R ) )  <->  [. 1o  /  n ]. ps )
15 biid 227 . 2  |-  ( [. 1o  /  n ]. th  <->  [. 1o  /  n ]. th )
16 biid 227 . 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 ) ) ) ) )
17 biid 227 . 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 ) ) ) ) )
18 biid 227 . . . . 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 ) ) )
196, 18, 7, 10bnj121 28902 . . . 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 ) ) )
208anbi2i 675 . . . . . . 7  |-  ( ( f  Fn  1o  /\  [. 1o  /  n ]. ph )  <->  ( f  Fn  1o  /\  ( f `
 (/) )  =  pred ( x ,  A ,  R ) ) )
2120, 13anbi12i 678 . . . . . 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 ) ) ) )
22 df-3an 936 . . . . . 6  |-  ( ( f  Fn  1o  /\  [. 1o  /  n ]. ph 
/\  [. 1o  /  n ]. ps )  <->  ( (
f  Fn  1o  /\  [. 1o  /  n ]. ph )  /\  [. 1o  /  n ]. ps )
)
23 df-3an 936 . . . . . 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 ) ) ) )
2421, 22, 233bitr4i 268 . . . . 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 ) ) ) )
2524imbi2i 303 . . . 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 ) ) ) ) )
2619, 25bitri 240 . . 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 ) ) ) ) )
2726bicomi 193 . 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 ) ) )
28 eqid 2283 . 2  |-  { <. (/)
,  pred ( x ,  A ,  R )
>. }  =  { <. (/)
,  pred ( x ,  A ,  R )
>. }
29 biid 227 . 2  |-  ( [. { <. (/) ,  pred (
x ,  A ,  R ) >. }  / 
f ]. ( f `  (/) )  =  pred (
x ,  A ,  R )  <->  [. { <. (/)
,  pred ( x ,  A ,  R )
>. }  /  f ]. ( f `  (/) )  = 
pred ( x ,  A ,  R ) )
30 biid 227 . 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 ) ) )
31 snex 4216 . . . 4  |-  { <. (/)
,  pred ( x ,  A ,  R )
>. }  e.  _V
3227, 31bnj524 28766 . . 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 ) ) )
33 biid 227 . . . . 5  |-  ( [. { <. (/) ,  pred (
x ,  A ,  R ) >. }  / 
f ]. [. 1o  /  n ]. ph  <->  [. { <. (/)
,  pred ( x ,  A ,  R )
>. }  /  f ]. [. 1o  /  n ]. ph )
34 biid 227 . . . . 5  |-  ( [. { <. (/) ,  pred (
x ,  A ,  R ) >. }  / 
f ]. [. 1o  /  n ]. ps  <->  [. { <. (/)
,  pred ( x ,  A ,  R )
>. }  /  f ]. [. 1o  /  n ]. ps )
35 biid 227 . . . . 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 ) ) )
3628, 33, 34, 35, 19bnj124 28903 . . . 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 ) ) )
371, 7, 33, 28bnj125 28904 . . . . . . . 8  |-  ( [. { <. (/) ,  pred (
x ,  A ,  R ) >. }  / 
f ]. [. 1o  /  n ]. ph  <->  ( { <.
(/) ,  pred ( x ,  A ,  R
) >. } `  (/) )  = 
pred ( x ,  A ,  R ) )
3837anbi2i 675 . . . . . . 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 ) ) )
392, 10, 34, 28bnj126 28905 . . . . . . 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 ) ) )
4038, 39anbi12i 678 . . . . . 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 ) ) ) )
41 df-3an 936 . . . . . 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 ) )
42 df-3an 936 . . . . . 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 ) ) ) )
4340, 41, 423bitr4i 268 . . . . 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 ) ) ) )
4443imbi2i 303 . . . 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 ) ) ) ) )
4536, 44bitri 240 . . 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 ) ) ) ) )
4632, 45bitr2i 241 . 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 ) ) ) ) )
47 biid 227 . 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 ) ) ) )
48 biid 227 . . . . 5  |-  ( ( f  Fn  1o  /\  [. 1o  /  n ]. ph 
/\  [. 1o  /  n ]. ps )  <->  ( f  Fn  1o  /\  [. 1o  /  n ]. ph  /\  [. 1o  /  n ]. ps ) )
49 biid 227 . . . . 5  |-  ( [. g  /  f ]. (
f  Fn  1o  /\  [. 1o  /  n ]. ph 
/\  [. 1o  /  n ]. ps )  <->  [. g  / 
f ]. ( f  Fn  1o  /\  [. 1o  /  n ]. ph  /\  [. 1o  /  n ]. ps ) )
50 biid 227 . . . . 5  |-  ( [. g  /  f ]. [. 1o  /  n ]. ph  <->  [. g  / 
f ]. [. 1o  /  n ]. ph )
51 biid 227 . . . . 5  |-  ( [. g  /  f ]. [. 1o  /  n ]. ps  <->  [. g  / 
f ]. [. 1o  /  n ]. ps )
5248, 49, 50, 51bnj156 28756 . . . 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 ) )
5350, 8bnj154 28910 . . . . . . 7  |-  ( [. g  /  f ]. [. 1o  /  n ]. ph  <->  ( g `  (/) )  =  pred ( x ,  A ,  R ) )
5453anbi2i 675 . . . . . 6  |-  ( ( g  Fn  1o  /\  [. g  /  f ]. [. 1o  /  n ]. ph )  <->  ( g  Fn  1o  /\  ( g `
 (/) )  =  pred ( x ,  A ,  R ) ) )
5551, 13bnj155 28911 . . . . . 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 ) ) )
5654, 55anbi12i 678 . . . . 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 ) ) ) )
57 df-3an 936 . . . . 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 )
)
58 df-3an 936 . . . . 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 ) ) ) )
5956, 57, 583bitr4i 268 . . . 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 ) ) ) )
6052, 59bitri 240 . . 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 ) ) ) )
6120, 12anbi12i 678 . . . . 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 ) ) ) )
6261, 22, 233bitr4i 268 . . . 4  |-  ( ( 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 ) ) ) )
6362sbcbii 3046 . . 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 ) ) ) )
6460, 63bitr3i 242 . 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 ) ) ) )
65 biid 227 . 2  |-  ( [. g  /  f ]. (
f `  (/) )  = 
pred ( x ,  A ,  R )  <->  [. g  /  f ]. ( f `  (/) )  = 
pred ( x ,  A ,  R ) )
66 biid 227 . 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 ) ) )
671, 2, 3, 4, 5, 6, 9, 14, 15, 16, 17, 27, 28, 29, 30, 46, 47, 64, 65, 66bnj151 28909 1  |-  ( n  =  1o  ->  (
( n  e.  D  /\  ta )  ->  th )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934   E.wex 1528    = wceq 1623    e. wcel 1684   E!weu 2143   E*wmo 2144   A.wral 2543   [.wsbc 2991    \ cdif 3149   (/)c0 3455   {csn 3640   <.cop 3643   U_ciun 3905   class class class wbr 4023    _E cep 4303   suc csuc 4394   omcom 4656    Fn wfn 5250   ` cfv 5255   1oc1o 6472    predc-bnj14 28713    FrSe w-bnj15 28717
This theorem is referenced by:  bnj852  28953
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-suc 4398  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-1o 6479  df-bnj13 28716  df-bnj15 28718
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