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Theorem wfr3g 27636
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
Dummy variables  w  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 r19.26 2847 . . . . . . 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 5688 . . . . . . . . . . . 12  |-  ( z  =  w  ->  ( F `  z )  =  ( F `  w ) )
3 fveq2 5688 . . . . . . . . . . . 12  |-  ( z  =  w  ->  ( G `  z )  =  ( G `  w ) )
42, 3eqeq12d 2455 . . . . . . . . . . 11  |-  ( z  =  w  ->  (
( F `  z
)  =  ( G `
 z )  <->  ( F `  w )  =  ( G `  w ) ) )
54imbi2d 316 . . . . . . . . . 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 1678 . . . . . . . . . . . 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 )
) ) ) )
76ra4 3279 . . . . . . . . . . 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 5688 . . . . . . . . . . . . . . . . . . 19  |-  ( y  =  z  ->  ( F `  y )  =  ( F `  z ) )
9 predeq3 27542 . . . . . . . . . . . . . . . . . . . . 21  |-  ( y  =  z  ->  Pred ( R ,  A , 
y )  =  Pred ( R ,  A , 
z ) )
109reseq2d 5106 . . . . . . . . . . . . . . . . . . . 20  |-  ( y  =  z  ->  ( F  |`  Pred ( R ,  A ,  y )
)  =  ( F  |`  Pred ( R ,  A ,  z )
) )
1110fveq2d 5692 . . . . . . . . . . . . . . . . . . 19  |-  ( y  =  z  ->  ( H `  ( F  |` 
Pred ( R ,  A ,  y )
) )  =  ( H `  ( F  |`  Pred ( R ,  A ,  z )
) ) )
128, 11eqeq12d 2455 . . . . . . . . . . . . . . . . . 18  |-  ( y  =  z  ->  (
( F `  y
)  =  ( H `
 ( F  |`  Pred ( R ,  A ,  y ) ) )  <->  ( F `  z )  =  ( H `  ( F  |`  Pred ( R ,  A ,  z )
) ) ) )
13 fveq2 5688 . . . . . . . . . . . . . . . . . . 19  |-  ( y  =  z  ->  ( G `  y )  =  ( G `  z ) )
149reseq2d 5106 . . . . . . . . . . . . . . . . . . . 20  |-  ( y  =  z  ->  ( G  |`  Pred ( R ,  A ,  y )
)  =  ( G  |`  Pred ( R ,  A ,  z )
) )
1514fveq2d 5692 . . . . . . . . . . . . . . . . . . 19  |-  ( y  =  z  ->  ( H `  ( G  |` 
Pred ( R ,  A ,  y )
) )  =  ( H `  ( G  |`  Pred ( R ,  A ,  z )
) ) )
1613, 15eqeq12d 2455 . . . . . . . . . . . . . . . . . 18  |-  ( y  =  z  ->  (
( G `  y
)  =  ( H `
 ( G  |`  Pred ( R ,  A ,  y ) ) )  <->  ( G `  z )  =  ( H `  ( G  |`  Pred ( R ,  A ,  z )
) ) ) )
1712, 16anbi12d 705 . . . . . . . . . . . . . . . . 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 ) ) ) ) ) )
1817rspcva 3068 . . . . . . . . . . . . . . . 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 27545 . . . . . . . . . . . . . . . . . . . . . . 23  |-  Pred ( R ,  A , 
z )  C_  A
20 fvreseq 5802 . . . . . . . . . . . . . . . . . . . . . . 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 666 . . . . . . . . . . . . . . . . . . . . . 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 482 . . . . . . . . . . . . . . . . . . . . 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 2446 . . . . . . . . . . . . . . . . . . . 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 5692 . . . . . . . . . . . . . . . . . . 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 2460 . . . . . . . . . . . . . . . . . . . . . 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 450 . . . . . . . . . . . . . . . . . . . . 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 2460 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( F `  z
)  =  ( H `
 ( G  |`  Pred ( R ,  A ,  z ) ) )  /\  ( G `
 z )  =  ( H `  ( G  |`  Pred ( R ,  A ,  z )
) ) )  -> 
( F `  z
)  =  ( G `
 z ) )
2827ex 434 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( F `  z )  =  ( H `  ( G  |`  Pred ( R ,  A , 
z ) ) )  ->  ( ( G `
 z )  =  ( H `  ( G  |`  Pred ( R ,  A ,  z )
) )  ->  ( F `  z )  =  ( G `  z ) ) )
2926, 28syl 16 . . . . . . . . . . . . . . . . . . . 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 600 . . . . . . . . . . . . . . . . . . 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 16 . . . . . . . . . . . . . . . . . 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 31 . . . . . . . . . . . . . . . . 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 436 . . . . . . . . . . . . . . . 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 16 . . . . . . . . . . . . . . 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 434 . . . . . . . . . . . . . 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 78 . . . . . . . . . . . . 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 431 . . . . . . . . . . . 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 26 . . . . . . . . . . 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 32 . . . . . . . . . 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 27587 . . . . . . . . 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 2801 . . . . . . . . 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 196 . . . . . . . 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 31 . . . . . . 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 473 . . . . . 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 817 . . . . 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 31 . . . 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 1180 . . 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 2441 . . 3  |-  A  =  A
4947, 48jctil 534 . 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 5795 . . . 4  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( F  =  G  <-> 
( A  =  A  /\  A. z  e.  A  ( F `  z )  =  ( G `  z ) ) ) )
5150ad2ant2r 741 . . 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 1001 . 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 232 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 960    = wceq 1364    e. wcel 1761   A.wral 2713    C_ wss 3325   Se wse 4673    We wwe 4674    |` cres 4838    Fn wfn 5410   ` cfv 5415   Predcpred 27537
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 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-fv 5423  df-pred 27538
This theorem is referenced by:  wfrlem5  27641  wfr3  27655
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