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Theorem wfr3g 7039
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 2919 . . . . . . 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 5870 . . . . . . . . . . . 12  |-  ( z  =  w  ->  ( F `  z )  =  ( F `  w ) )
3 fveq2 5870 . . . . . . . . . . . 12  |-  ( z  =  w  ->  ( G `  z )  =  ( G `  w ) )
42, 3eqeq12d 2468 . . . . . . . . . . 11  |-  ( z  =  w  ->  (
( F `  z
)  =  ( G `
 z )  <->  ( F `  w )  =  ( G `  w ) ) )
54imbi2d 318 . . . . . . . . . 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 ra4v 3354 . . . . . . . . . . 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
) ) )
7 fveq2 5870 . . . . . . . . . . . . . . . . . . 19  |-  ( y  =  z  ->  ( F `  y )  =  ( F `  z ) )
8 predeq3 5387 . . . . . . . . . . . . . . . . . . . . 21  |-  ( y  =  z  ->  Pred ( R ,  A , 
y )  =  Pred ( R ,  A , 
z ) )
98reseq2d 5108 . . . . . . . . . . . . . . . . . . . 20  |-  ( y  =  z  ->  ( F  |`  Pred ( R ,  A ,  y )
)  =  ( F  |`  Pred ( R ,  A ,  z )
) )
109fveq2d 5874 . . . . . . . . . . . . . . . . . . 19  |-  ( y  =  z  ->  ( H `  ( F  |` 
Pred ( R ,  A ,  y )
) )  =  ( H `  ( F  |`  Pred ( R ,  A ,  z )
) ) )
117, 10eqeq12d 2468 . . . . . . . . . . . . . . . . . 18  |-  ( y  =  z  ->  (
( F `  y
)  =  ( H `
 ( F  |`  Pred ( R ,  A ,  y ) ) )  <->  ( F `  z )  =  ( H `  ( F  |`  Pred ( R ,  A ,  z )
) ) ) )
12 fveq2 5870 . . . . . . . . . . . . . . . . . . 19  |-  ( y  =  z  ->  ( G `  y )  =  ( G `  z ) )
138reseq2d 5108 . . . . . . . . . . . . . . . . . . . 20  |-  ( y  =  z  ->  ( G  |`  Pred ( R ,  A ,  y )
)  =  ( G  |`  Pred ( R ,  A ,  z )
) )
1413fveq2d 5874 . . . . . . . . . . . . . . . . . . 19  |-  ( y  =  z  ->  ( H `  ( G  |` 
Pred ( R ,  A ,  y )
) )  =  ( H `  ( G  |`  Pred ( R ,  A ,  z )
) ) )
1512, 14eqeq12d 2468 . . . . . . . . . . . . . . . . . 18  |-  ( y  =  z  ->  (
( G `  y
)  =  ( H `
 ( G  |`  Pred ( R ,  A ,  y ) ) )  <->  ( G `  z )  =  ( H `  ( G  |`  Pred ( R ,  A ,  z )
) ) ) )
1611, 15anbi12d 718 . . . . . . . . . . . . . . . . 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 ) ) ) ) ) )
1716rspcva 3150 . . . . . . . . . . . . . . . 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 ) ) ) ) )
18 predss 5390 . . . . . . . . . . . . . . . . . . . . . . 23  |-  Pred ( R ,  A , 
z )  C_  A
19 fvreseq 5989 . . . . . . . . . . . . . . . . . . . . . . 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
) ) )
2018, 19mpan2 678 . . . . . . . . . . . . . . . . . . . . . 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
) ) )
2120biimpar 488 . . . . . . . . . . . . . . . . . . . . 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 ) ) )
2221eqcomd 2459 . . . . . . . . . . . . . . . . . . . 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 ) ) )
2322fveq2d 5874 . . . . . . . . . . . . . . . . . . 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 ) ) ) )
24 eqtr3 2474 . . . . . . . . . . . . . . . . . . . . . 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 ) ) ) )
2524ancoms 455 . . . . . . . . . . . . . . . . . . . . 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 )
) ) )
26 eqtr3 2474 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( F `  z
)  =  ( H `
 ( G  |`  Pred ( R ,  A ,  z ) ) )  /\  ( G `
 z )  =  ( H `  ( G  |`  Pred ( R ,  A ,  z )
) ) )  -> 
( F `  z
)  =  ( G `
 z ) )
2726ex 436 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( F `  z )  =  ( H `  ( G  |`  Pred ( R ,  A , 
z ) ) )  ->  ( ( G `
 z )  =  ( H `  ( G  |`  Pred ( R ,  A ,  z )
) )  ->  ( F `  z )  =  ( G `  z ) ) )
2825, 27syl 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 ) ) )
2928expimpd 608 . . . . . . . . . . . . . . . . . . 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 ) ) )
3023, 29syl 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 ) ) )
3130com12 32 . . . . . . . . . . . . . . . . 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 ) ) )
3231expd 438 . . . . . . . . . . . . . . . 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 ) ) ) )
3317, 32syl 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 ) ) ) )
3433ex 436 . . . . . . . . . . . . . 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 ) ) ) ) )
3534com23 81 . . . . . . . . . . . . 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 ) ) ) ) )
3635impd 433 . . . . . . . . . . . 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 ) ) ) )
3736a2d 29 . . . . . . . . . . 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 ) ) ) )
386, 37syl5 33 . . . . . . . . . 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 ) ) ) )
395, 38wfis2g 5422 . . . . . . . . 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 ) ) )
40 r19.21v 2795 . . . . . . . . 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 ) ) )
4139, 40sylib 200 . . . . . . . 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 ) ) )
4241com12 32 . . . . . . 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 ) ) )
431, 42sylan2br 479 . . . . . 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 ) ) )
4443an4s 836 . . . . 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 ) ) )
4544com12 32 . . . 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 ) ) )
46453impib 1207 . . 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 ) )
47 eqid 2453 . . 3  |-  A  =  A
4846, 47jctil 540 . 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 ) ) )
49 eqfnfv2 5982 . . . 4  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( F  =  G  <-> 
( A  =  A  /\  A. z  e.  A  ( F `  z )  =  ( G `  z ) ) ) )
5049ad2ant2r 754 . . 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 ) ) ) )
51503adant1 1027 . 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 ) ) ) )
5248, 51mpbird 236 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 188    /\ wa 371    /\ w3a 986    = wceq 1446    e. wcel 1889   A.wral 2739    C_ wss 3406   Se wse 4794    We wwe 4795    |` cres 4839   Predcpred 5382    Fn wfn 5580   ` cfv 5585
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-reu 2746  df-rmo 2747  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3734  df-if 3884  df-sn 3971  df-pr 3973  df-op 3977  df-uni 4202  df-br 4406  df-opab 4465  df-mpt 4466  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-se 4797  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 5383  df-iota 5549  df-fun 5587  df-fn 5588  df-fv 5593
This theorem is referenced by:  wfrlem5  7045  wfr3  7061
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