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Theorem wfr3g 25469
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 2798 . . . . . . 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 5687 . . . . . . . . . . . 12  |-  ( z  =  w  ->  ( F `  z )  =  ( F `  w ) )
3 fveq2 5687 . . . . . . . . . . . 12  |-  ( z  =  w  ->  ( G `  z )  =  ( G `  w ) )
42, 3eqeq12d 2418 . . . . . . . . . . 11  |-  ( z  =  w  ->  (
( F `  z
)  =  ( G `
 z )  <->  ( F `  w )  =  ( G `  w ) ) )
54imbi2d 308 . . . . . . . . . 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 1626 . . . . . . . . . . . 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 3205 . . . . . . . . . . 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 5687 . . . . . . . . . . . . . . . . . . 19  |-  ( y  =  z  ->  ( F `  y )  =  ( F `  z ) )
9 predeq3 25385 . . . . . . . . . . . . . . . . . . . . 21  |-  ( y  =  z  ->  Pred ( R ,  A , 
y )  =  Pred ( R ,  A , 
z ) )
109reseq2d 5105 . . . . . . . . . . . . . . . . . . . 20  |-  ( y  =  z  ->  ( F  |`  Pred ( R ,  A ,  y )
)  =  ( F  |`  Pred ( R ,  A ,  z )
) )
1110fveq2d 5691 . . . . . . . . . . . . . . . . . . 19  |-  ( y  =  z  ->  ( H `  ( F  |` 
Pred ( R ,  A ,  y )
) )  =  ( H `  ( F  |`  Pred ( R ,  A ,  z )
) ) )
128, 11eqeq12d 2418 . . . . . . . . . . . . . . . . . 18  |-  ( y  =  z  ->  (
( F `  y
)  =  ( H `
 ( F  |`  Pred ( R ,  A ,  y ) ) )  <->  ( F `  z )  =  ( H `  ( F  |`  Pred ( R ,  A ,  z )
) ) ) )
13 fveq2 5687 . . . . . . . . . . . . . . . . . . 19  |-  ( y  =  z  ->  ( G `  y )  =  ( G `  z ) )
149reseq2d 5105 . . . . . . . . . . . . . . . . . . . 20  |-  ( y  =  z  ->  ( G  |`  Pred ( R ,  A ,  y )
)  =  ( G  |`  Pred ( R ,  A ,  z )
) )
1514fveq2d 5691 . . . . . . . . . . . . . . . . . . 19  |-  ( y  =  z  ->  ( H `  ( G  |` 
Pred ( R ,  A ,  y )
) )  =  ( H `  ( G  |`  Pred ( R ,  A ,  z )
) ) )
1613, 15eqeq12d 2418 . . . . . . . . . . . . . . . . . 18  |-  ( y  =  z  ->  (
( G `  y
)  =  ( H `
 ( G  |`  Pred ( R ,  A ,  y ) ) )  <->  ( G `  z )  =  ( H `  ( G  |`  Pred ( R ,  A ,  z )
) ) ) )
1712, 16anbi12d 692 . . . . . . . . . . . . . . . . 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 3010 . . . . . . . . . . . . . . . 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 25387 . . . . . . . . . . . . . . . . . . . . . . 23  |-  Pred ( R ,  A , 
z )  C_  A
20 fvreseq 5792 . . . . . . . . . . . . . . . . . . . . . . 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 653 . . . . . . . . . . . . . . . . . . . . . 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 472 . . . . . . . . . . . . . . . . . . . . 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 2409 . . . . . . . . . . . . . . . . . . . 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 5691 . . . . . . . . . . . . . . . . . . 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 2423 . . . . . . . . . . . . . . . . . . . . . 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 440 . . . . . . . . . . . . . . . . . . . . 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 2423 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( F `  z
)  =  ( H `
 ( G  |`  Pred ( R ,  A ,  z ) ) )  /\  ( G `
 z )  =  ( H `  ( G  |`  Pred ( R ,  A ,  z )
) ) )  -> 
( F `  z
)  =  ( G `
 z ) )
2827ex 424 . . . . . . . . . . . . . . . . . . . . 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 587 . . . . . . . . . . . . . . . . . . 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 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 426 . . . . . . . . . . . . . . . 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 424 . . . . . . . . . . . . . 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 421 . . . . . . . . . . . 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 24 . . . . . . . . . . 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 25427 . . . . . . . . 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 2753 . . . . . . . . 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 189 . . . . . . . 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 463 . . . . . 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 800 . . . . 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 1151 . . 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 2404 . . 3  |-  A  =  A
4947, 48jctil 524 . 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 5787 . . . 4  |-  ( ( F  Fn  A  /\  G  Fn  A )  ->  ( F  =  G  <-> 
( A  =  A  /\  A. z  e.  A  ( F `  z )  =  ( G `  z ) ) ) )
5150ad2ant2r 728 . . 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 975 . 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 224 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 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   A.wral 2666    C_ wss 3280   Se wse 4499    We wwe 4500    |` cres 4839    Fn wfn 5408   ` cfv 5413   Predcpred 25381
This theorem is referenced by:  wfrlem5  25474  wfr3  25489
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  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-iota 5377  df-fun 5415  df-fn 5416  df-fv 5421  df-pred 25382
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