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Theorem wfr3g 23624
Description: Functions defined by well founded recursion are identical up to relation, domain, and characteristic function. (Contributed by Scott Fenton, 11-Feb-2011.)
Assertion
Ref Expression
wfr3g  |-  ( ( ( R  We  A  /\  R Se  A )  /\  ( F  Fn  A  /\  A. y  e.  A  ( F `  y )  =  ( H `  ( F  |`  Pred ( R ,  A , 
y ) ) ) )  /\  ( G  Fn  A  /\  A. y  e.  A  ( G `  y )  =  ( H `  ( G  |`  Pred ( R ,  A , 
y ) ) ) ) )  ->  F  =  G )
Distinct variable groups:    y, A    y, F    y, G    y, H    y, R

Proof of Theorem wfr3g
StepHypRef Expression
1 r19.26 2650 . . . . . . 7  |-  ( A. y  e.  A  (
( F `  y
)  =  ( H `
 ( F  |`  Pred ( R ,  A ,  y ) ) )  /\  ( G `
 y )  =  ( H `  ( G  |`  Pred ( R ,  A ,  y )
) ) )  <->  ( A. y  e.  A  ( F `  y )  =  ( H `  ( F  |`  Pred ( R ,  A , 
y ) ) )  /\  A. y  e.  A  ( G `  y )  =  ( H `  ( G  |`  Pred ( R ,  A ,  y )
) ) ) )
2 fveq2 5458 . . . . . . . . . . . 12  |-  ( z  =  w  ->  ( F `  z )  =  ( F `  w ) )
3 fveq2 5458 . . . . . . . . . . . 12  |-  ( z  =  w  ->  ( G `  z )  =  ( G `  w ) )
42, 3eqeq12d 2272 . . . . . . . . . . 11  |-  ( z  =  w  ->  (
( F `  z
)  =  ( G `
 z )  <->  ( F `  w )  =  ( G `  w ) ) )
54imbi2d 309 . . . . . . . . . 10  |-  ( z  =  w  ->  (
( ( ( F  Fn  A  /\  G  Fn  A )  /\  A. y  e.  A  (
( F `  y
)  =  ( H `
 ( F  |`  Pred ( R ,  A ,  y ) ) )  /\  ( G `
 y )  =  ( H `  ( G  |`  Pred ( R ,  A ,  y )
) ) ) )  ->  ( F `  z )  =  ( G `  z ) )  <->  ( ( ( F  Fn  A  /\  G  Fn  A )  /\  A. y  e.  A  ( ( F `  y )  =  ( H `  ( F  |`  Pred ( R ,  A ,  y )
) )  /\  ( G `  y )  =  ( H `  ( G  |`  Pred ( R ,  A , 
y ) ) ) ) )  ->  ( F `  w )  =  ( G `  w ) ) ) )
6 nfv 1629 . . . . . . . . . . . 12  |-  F/ w
( ( F  Fn  A  /\  G  Fn  A
)  /\  A. y  e.  A  ( ( F `  y )  =  ( H `  ( F  |`  Pred ( R ,  A , 
y ) ) )  /\  ( G `  y )  =  ( H `  ( G  |`  Pred ( R ,  A ,  y )
) ) ) )
76ra5 3050 . . . . . . . . . . 11  |-  ( A. w  e.  Pred  ( R ,  A ,  z ) ( ( ( F  Fn  A  /\  G  Fn  A )  /\  A. y  e.  A  ( ( F `  y )  =  ( H `  ( F  |`  Pred ( R ,  A ,  y )
) )  /\  ( G `  y )  =  ( H `  ( G  |`  Pred ( R ,  A , 
y ) ) ) ) )  ->  ( F `  w )  =  ( G `  w ) )  -> 
( ( ( F  Fn  A  /\  G  Fn  A )  /\  A. y  e.  A  (
( F `  y
)  =  ( H `
 ( F  |`  Pred ( R ,  A ,  y ) ) )  /\  ( G `
 y )  =  ( H `  ( G  |`  Pred ( R ,  A ,  y )
) ) ) )  ->  A. w  e.  Pred  ( R ,  A , 
z ) ( F `
 w )  =  ( G `  w
) ) )
8 fveq2 5458 . . . . . . . . . . . . . . . . . . 19  |-  ( y  =  z  ->  ( F `  y )  =  ( F `  z ) )
9 predeq3 23540 . . . . . . . . . . . . . . . . . . . . 21  |-  ( y  =  z  ->  Pred ( R ,  A , 
y )  =  Pred ( R ,  A , 
z ) )
109reseq2d 4943 . . . . . . . . . . . . . . . . . . . 20  |-  ( y  =  z  ->  ( F  |`  Pred ( R ,  A ,  y )
)  =  ( F  |`  Pred ( R ,  A ,  z )
) )
1110fveq2d 5462 . . . . . . . . . . . . . . . . . . 19  |-  ( y  =  z  ->  ( H `  ( F  |` 
Pred ( R ,  A ,  y )
) )  =  ( H `  ( F  |`  Pred ( R ,  A ,  z )
) ) )
128, 11eqeq12d 2272 . . . . . . . . . . . . . . . . . 18  |-  ( y  =  z  ->  (
( F `  y
)  =  ( H `
 ( F  |`  Pred ( R ,  A ,  y ) ) )  <->  ( F `  z )  =  ( H `  ( F  |`  Pred ( R ,  A ,  z )
) ) ) )
13 fveq2 5458 . . . . . . . . . . . . . . . . . . 19  |-  ( y  =  z  ->  ( G `  y )  =  ( G `  z ) )
149reseq2d 4943 . . . . . . . . . . . . . . . . . . . 20  |-  ( y  =  z  ->  ( G  |`  Pred ( R ,  A ,  y )
)  =  ( G  |`  Pred ( R ,  A ,  z )
) )
1514fveq2d 5462 . . . . . . . . . . . . . . . . . . 19  |-  ( y  =  z  ->  ( H `  ( G  |` 
Pred ( R ,  A ,  y )
) )  =  ( H `  ( G  |`  Pred ( R ,  A ,  z )
) ) )
1613, 15eqeq12d 2272 . . . . . . . . . . . . . . . . . 18  |-  ( y  =  z  ->  (
( G `  y
)  =  ( H `
 ( G  |`  Pred ( R ,  A ,  y ) ) )  <->  ( G `  z )  =  ( H `  ( G  |`  Pred ( R ,  A ,  z )
) ) ) )
1712, 16anbi12d 694 . . . . . . . . . . . . . . . . 17  |-  ( y  =  z  ->  (
( ( F `  y )  =  ( H `  ( F  |`  Pred ( R ,  A ,  y )
) )  /\  ( G `  y )  =  ( H `  ( G  |`  Pred ( R ,  A , 
y ) ) ) )  <->  ( ( F `
 z )  =  ( H `  ( F  |`  Pred ( R ,  A ,  z )
) )  /\  ( G `  z )  =  ( H `  ( G  |`  Pred ( R ,  A , 
z ) ) ) ) ) )
1817rcla4va 2857 . . . . . . . . . . . . . . . 16  |-  ( ( z  e.  A  /\  A. y  e.  A  ( ( F `  y
)  =  ( H `
 ( F  |`  Pred ( R ,  A ,  y ) ) )  /\  ( G `
 y )  =  ( H `  ( G  |`  Pred ( R ,  A ,  y )
) ) ) )  ->  ( ( F `
 z )  =  ( H `  ( F  |`  Pred ( R ,  A ,  z )
) )  /\  ( G `  z )  =  ( H `  ( G  |`  Pred ( R ,  A , 
z ) ) ) ) )
19 predss 23542 . . . . . . . . . . . . . . . . . . . . . . 23  |-  Pred ( R ,  A , 
z )  C_  A
20 fvreseq 5562 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  Pred ( R ,  A ,  z )  C_  A )  ->  ( ( F  |`  Pred ( R ,  A ,  z ) )  =  ( G  |`  Pred ( R ,  A ,  z ) )  <->  A. w  e.  Pred  ( R ,  A , 
z ) ( F `
 w )  =  ( G `  w
) ) )
2119, 20mpan2 655 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( ( F  |`  Pred ( R ,  A ,  z ) )  =  ( G  |`  Pred ( R ,  A ,  z ) )  <->  A. w  e.  Pred  ( R ,  A , 
z ) ( F `
 w )  =  ( G `  w
) ) )
2221biimpar 473 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  A. w  e.  Pred  ( R ,  A ,  z )
( F `  w
)  =  ( G `
 w ) )  ->  ( F  |`  Pred ( R ,  A ,  z ) )  =  ( G  |`  Pred ( R ,  A ,  z ) ) )
2322eqcomd 2263 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  A. w  e.  Pred  ( R ,  A ,  z )
( F `  w
)  =  ( G `
 w ) )  ->  ( G  |`  Pred ( R ,  A ,  z ) )  =  ( F  |`  Pred ( R ,  A ,  z ) ) )
2423fveq2d 5462 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  A. w  e.  Pred  ( R ,  A ,  z )
( F `  w
)  =  ( G `
 w ) )  ->  ( H `  ( G  |`  Pred ( R ,  A , 
z ) ) )  =  ( H `  ( F  |`  Pred ( R ,  A , 
z ) ) ) )
25 eqtr3 2277 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( F `  z
)  =  ( H `
 ( F  |`  Pred ( R ,  A ,  z ) ) )  /\  ( H `
 ( G  |`  Pred ( R ,  A ,  z ) ) )  =  ( H `
 ( F  |`  Pred ( R ,  A ,  z ) ) ) )  ->  ( F `  z )  =  ( H `  ( G  |`  Pred ( R ,  A , 
z ) ) ) )
2625ancoms 441 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( H `  ( G  |`  Pred ( R ,  A ,  z )
) )  =  ( H `  ( F  |`  Pred ( R ,  A ,  z )
) )  /\  ( F `  z )  =  ( H `  ( F  |`  Pred ( R ,  A , 
z ) ) ) )  ->  ( F `  z )  =  ( H `  ( G  |`  Pred ( R ,  A ,  z )
) ) )
27 eqtr3 2277 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( F `  z
)  =  ( H `
 ( G  |`  Pred ( R ,  A ,  z ) ) )  /\  ( G `
 z )  =  ( H `  ( G  |`  Pred ( R ,  A ,  z )
) ) )  -> 
( F `  z
)  =  ( G `
 z ) )
2827ex 425 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( F `  z )  =  ( H `  ( G  |`  Pred ( R ,  A , 
z ) ) )  ->  ( ( G `
 z )  =  ( H `  ( G  |`  Pred ( R ,  A ,  z )
) )  ->  ( F `  z )  =  ( G `  z ) ) )
2926, 28syl 17 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( H `  ( G  |`  Pred ( R ,  A ,  z )
) )  =  ( H `  ( F  |`  Pred ( R ,  A ,  z )
) )  /\  ( F `  z )  =  ( H `  ( F  |`  Pred ( R ,  A , 
z ) ) ) )  ->  ( ( G `  z )  =  ( H `  ( G  |`  Pred ( R ,  A , 
z ) ) )  ->  ( F `  z )  =  ( G `  z ) ) )
3029expimpd 589 . . . . . . . . . . . . . . . . . . 19  |-  ( ( H `  ( G  |`  Pred ( R ,  A ,  z )
) )  =  ( H `  ( F  |`  Pred ( R ,  A ,  z )
) )  ->  (
( ( F `  z )  =  ( H `  ( F  |`  Pred ( R ,  A ,  z )
) )  /\  ( G `  z )  =  ( H `  ( G  |`  Pred ( R ,  A , 
z ) ) ) )  ->  ( F `  z )  =  ( G `  z ) ) )
3124, 30syl 17 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  A. w  e.  Pred  ( R ,  A ,  z )
( F `  w
)  =  ( G `
 w ) )  ->  ( ( ( F `  z )  =  ( H `  ( F  |`  Pred ( R ,  A , 
z ) ) )  /\  ( G `  z )  =  ( H `  ( G  |`  Pred ( R ,  A ,  z )
) ) )  -> 
( F `  z
)  =  ( G `
 z ) ) )
3231com12 29 . . . . . . . . . . . . . . . . 17  |-  ( ( ( F `  z
)  =  ( H `
 ( F  |`  Pred ( R ,  A ,  z ) ) )  /\  ( G `
 z )  =  ( H `  ( G  |`  Pred ( R ,  A ,  z )
) ) )  -> 
( ( ( F  Fn  A  /\  G  Fn  A )  /\  A. w  e.  Pred  ( R ,  A ,  z ) ( F `  w )  =  ( G `  w ) )  ->  ( F `  z )  =  ( G `  z ) ) )
3332exp3a 427 . . . . . . . . . . . . . . . 16  |-  ( ( ( F `  z
)  =  ( H `
 ( F  |`  Pred ( R ,  A ,  z ) ) )  /\  ( G `
 z )  =  ( H `  ( G  |`  Pred ( R ,  A ,  z )
) ) )  -> 
( ( F  Fn  A  /\  G  Fn  A
)  ->  ( A. w  e.  Pred  ( R ,  A ,  z ) ( F `  w )  =  ( G `  w )  ->  ( F `  z )  =  ( G `  z ) ) ) )
3418, 33syl 17 . . . . . . . . . . . . . . 15  |-  ( ( z  e.  A  /\  A. y  e.  A  ( ( F `  y
)  =  ( H `
 ( F  |`  Pred ( R ,  A ,  y ) ) )  /\  ( G `
 y )  =  ( H `  ( G  |`  Pred ( R ,  A ,  y )
) ) ) )  ->  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( A. w  e.  Pred  ( R ,  A , 
z ) ( F `
 w )  =  ( G `  w
)  ->  ( F `  z )  =  ( G `  z ) ) ) )
3534ex 425 . . . . . . . . . . . . . 14  |-  ( z  e.  A  ->  ( A. y  e.  A  ( ( F `  y )  =  ( H `  ( F  |`  Pred ( R ,  A ,  y )
) )  /\  ( G `  y )  =  ( H `  ( G  |`  Pred ( R ,  A , 
y ) ) ) )  ->  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( A. w  e. 
Pred  ( R ,  A ,  z )
( F `  w
)  =  ( G `
 w )  -> 
( F `  z
)  =  ( G `
 z ) ) ) ) )
3635com23 74 . . . . . . . . . . . . 13  |-  ( z  e.  A  ->  (
( F  Fn  A  /\  G  Fn  A
)  ->  ( A. y  e.  A  (
( F `  y
)  =  ( H `
 ( F  |`  Pred ( R ,  A ,  y ) ) )  /\  ( G `
 y )  =  ( H `  ( G  |`  Pred ( R ,  A ,  y )
) ) )  -> 
( A. w  e. 
Pred  ( R ,  A ,  z )
( F `  w
)  =  ( G `
 w )  -> 
( F `  z
)  =  ( G `
 z ) ) ) ) )
3736imp3a 422 . . . . . . . . . . . 12  |-  ( z  e.  A  ->  (
( ( F  Fn  A  /\  G  Fn  A
)  /\  A. y  e.  A  ( ( F `  y )  =  ( H `  ( F  |`  Pred ( R ,  A , 
y ) ) )  /\  ( G `  y )  =  ( H `  ( G  |`  Pred ( R ,  A ,  y )
) ) ) )  ->  ( A. w  e.  Pred  ( R ,  A ,  z )
( F `  w
)  =  ( G `
 w )  -> 
( F `  z
)  =  ( G `
 z ) ) ) )
3837a2d 25 . . . . . . . . . . 11  |-  ( z  e.  A  ->  (
( ( ( F  Fn  A  /\  G  Fn  A )  /\  A. y  e.  A  (
( F `  y
)  =  ( H `
 ( F  |`  Pred ( R ,  A ,  y ) ) )  /\  ( G `
 y )  =  ( H `  ( G  |`  Pred ( R ,  A ,  y )
) ) ) )  ->  A. w  e.  Pred  ( R ,  A , 
z ) ( F `
 w )  =  ( G `  w
) )  ->  (
( ( F  Fn  A  /\  G  Fn  A
)  /\  A. y  e.  A  ( ( F `  y )  =  ( H `  ( F  |`  Pred ( R ,  A , 
y ) ) )  /\  ( G `  y )  =  ( H `  ( G  |`  Pred ( R ,  A ,  y )
) ) ) )  ->  ( F `  z )  =  ( G `  z ) ) ) )
397, 38syl5 30 . . . . . . . . . 10  |-  ( z  e.  A  ->  ( A. w  e.  Pred  ( R ,  A , 
z ) ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  A. y  e.  A  ( ( F `  y )  =  ( H `  ( F  |`  Pred ( R ,  A , 
y ) ) )  /\  ( G `  y )  =  ( H `  ( G  |`  Pred ( R ,  A ,  y )
) ) ) )  ->  ( F `  w )  =  ( G `  w ) )  ->  ( (
( F  Fn  A  /\  G  Fn  A
)  /\  A. y  e.  A  ( ( F `  y )  =  ( H `  ( F  |`  Pred ( R ,  A , 
y ) ) )  /\  ( G `  y )  =  ( H `  ( G  |`  Pred ( R ,  A ,  y )
) ) ) )  ->  ( F `  z )  =  ( G `  z ) ) ) )
405, 39wfis2g 23582 . . . . . . . . 9  |-  ( ( R  We  A  /\  R Se  A )  ->  A. z  e.  A  ( (
( F  Fn  A  /\  G  Fn  A
)  /\  A. y  e.  A  ( ( F `  y )  =  ( H `  ( F  |`  Pred ( R ,  A , 
y ) ) )  /\  ( G `  y )  =  ( H `  ( G  |`  Pred ( R ,  A ,  y )
) ) ) )  ->  ( F `  z )  =  ( G `  z ) ) )
41 r19.21v 2605 . . . . . . . . 9  |-  ( A. z  e.  A  (
( ( F  Fn  A  /\  G  Fn  A
)  /\  A. y  e.  A  ( ( F `  y )  =  ( H `  ( F  |`  Pred ( R ,  A , 
y ) ) )  /\  ( G `  y )  =  ( H `  ( G  |`  Pred ( R ,  A ,  y )
) ) ) )  ->  ( F `  z )  =  ( G `  z ) )  <->  ( ( ( F  Fn  A  /\  G  Fn  A )  /\  A. y  e.  A  ( ( F `  y )  =  ( H `  ( F  |`  Pred ( R ,  A ,  y )
) )  /\  ( G `  y )  =  ( H `  ( G  |`  Pred ( R ,  A , 
y ) ) ) ) )  ->  A. z  e.  A  ( F `  z )  =  ( G `  z ) ) )
4240, 41sylib 190 . . . . . . . 8  |-  ( ( R  We  A  /\  R Se  A )  ->  (
( ( F  Fn  A  /\  G  Fn  A
)  /\  A. y  e.  A  ( ( F `  y )  =  ( H `  ( F  |`  Pred ( R ,  A , 
y ) ) )  /\  ( G `  y )  =  ( H `  ( G  |`  Pred ( R ,  A ,  y )
) ) ) )  ->  A. z  e.  A  ( F `  z )  =  ( G `  z ) ) )
4342com12 29 . . . . . . 7  |-  ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  A. y  e.  A  ( ( F `  y )  =  ( H `  ( F  |`  Pred ( R ,  A , 
y ) ) )  /\  ( G `  y )  =  ( H `  ( G  |`  Pred ( R ,  A ,  y )
) ) ) )  ->  ( ( R  We  A  /\  R Se  A )  ->  A. z  e.  A  ( F `  z )  =  ( G `  z ) ) )
441, 43sylan2br 464 . . . . . 6  |-  ( ( ( F  Fn  A  /\  G  Fn  A
)  /\  ( A. y  e.  A  ( F `  y )  =  ( H `  ( F  |`  Pred ( R ,  A , 
y ) ) )  /\  A. y  e.  A  ( G `  y )  =  ( H `  ( G  |`  Pred ( R ,  A ,  y )
) ) ) )  ->  ( ( R  We  A  /\  R Se  A )  ->  A. z  e.  A  ( F `  z )  =  ( G `  z ) ) )
4544an4s 802 . . . . 5  |-  ( ( ( F  Fn  A  /\  A. y  e.  A  ( F `  y )  =  ( H `  ( F  |`  Pred ( R ,  A , 
y ) ) ) )  /\  ( G  Fn  A  /\  A. y  e.  A  ( G `  y )  =  ( H `  ( G  |`  Pred ( R ,  A , 
y ) ) ) ) )  ->  (
( R  We  A  /\  R Se  A )  ->  A. z  e.  A  ( F `  z )  =  ( G `  z ) ) )
4645com12 29 . . . 4  |-  ( ( R  We  A  /\  R Se  A )  ->  (
( ( F  Fn  A  /\  A. y  e.  A  ( F `  y )  =  ( H `  ( F  |`  Pred ( R ,  A ,  y )
) ) )  /\  ( G  Fn  A  /\  A. y  e.  A  ( G `  y )  =  ( H `  ( G  |`  Pred ( R ,  A , 
y ) ) ) ) )  ->  A. z  e.  A  ( F `  z )  =  ( G `  z ) ) )
47463impib 1154 . . 3  |-  ( ( ( R  We  A  /\  R Se  A )  /\  ( F  Fn  A  /\  A. y  e.  A  ( F `  y )  =  ( H `  ( F  |`  Pred ( R ,  A , 
y ) ) ) )  /\  ( G  Fn  A  /\  A. y  e.  A  ( G `  y )  =  ( H `  ( G  |`  Pred ( R ,  A , 
y ) ) ) ) )  ->  A. z  e.  A  ( F `  z )  =  ( G `  z ) )
48 eqid 2258 . . 3  |-  A  =  A
4947, 48jctil 525 . 2  |-  ( ( ( R  We  A  /\  R Se  A )  /\  ( F  Fn  A  /\  A. y  e.  A  ( F `  y )  =  ( H `  ( F  |`  Pred ( R ,  A , 
y ) ) ) )  /\  ( G  Fn  A  /\  A. y  e.  A  ( G `  y )  =  ( H `  ( G  |`  Pred ( R ,  A , 
y ) ) ) ) )  ->  ( A  =  A  /\  A. z  e.  A  ( F `  z )  =  ( G `  z ) ) )
50 eqfnfv2 5557 . . . 4  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( F  =  G  <-> 
( A  =  A  /\  A. z  e.  A  ( F `  z )  =  ( G `  z ) ) ) )
5150ad2ant2r 730 . . 3  |-  ( ( ( F  Fn  A  /\  A. y  e.  A  ( F `  y )  =  ( H `  ( F  |`  Pred ( R ,  A , 
y ) ) ) )  /\  ( G  Fn  A  /\  A. y  e.  A  ( G `  y )  =  ( H `  ( G  |`  Pred ( R ,  A , 
y ) ) ) ) )  ->  ( F  =  G  <->  ( A  =  A  /\  A. z  e.  A  ( F `  z )  =  ( G `  z ) ) ) )
52513adant1 978 . 2  |-  ( ( ( R  We  A  /\  R Se  A )  /\  ( F  Fn  A  /\  A. y  e.  A  ( F `  y )  =  ( H `  ( F  |`  Pred ( R ,  A , 
y ) ) ) )  /\  ( G  Fn  A  /\  A. y  e.  A  ( G `  y )  =  ( H `  ( G  |`  Pred ( R ,  A , 
y ) ) ) ) )  ->  ( F  =  G  <->  ( A  =  A  /\  A. z  e.  A  ( F `  z )  =  ( G `  z ) ) ) )
5349, 52mpbird 225 1  |-  ( ( ( R  We  A  /\  R Se  A )  /\  ( F  Fn  A  /\  A. y  e.  A  ( F `  y )  =  ( H `  ( F  |`  Pred ( R ,  A , 
y ) ) ) )  /\  ( G  Fn  A  /\  A. y  e.  A  ( G `  y )  =  ( H `  ( G  |`  Pred ( R ,  A , 
y ) ) ) ) )  ->  F  =  G )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621   A.wral 2518    C_ wss 3127   Se wse 4322    We wwe 4323    |` cres 4663    Fn wfn 4668   ` cfv 4673   Predcpred 23536
This theorem is referenced by:  wfrlem5  23629  wfr3  23644
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pr 4186  ax-un 4484
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-br 3998  df-opab 4052  df-mpt 4053  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-se 4325  df-we 4326  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-fv 4689  df-pred 23537
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