Theorem eqfnov 5870

Description: Equality of two operations is determined by their values. (Contributed by NM, 1-Sep-2005.)

Assertion

Ref	Expression
eqfnov	|- ( ( F Fn ( A X. B ) /\ G Fn ( C X. D ) ) -> ( F = G <-> ( ( A X. B ) = ( C X. D ) /\ A. x e. A A. y e. B ( x F y ) = ( x G y ) ) ) )

Distinct variable groups:   x, y, A   x, B, y   x, F, y   x, G, y

Allowed substitution hints:   C( x, y)   D( x, y)

Proof of Theorem eqfnov

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqfnfv2 5512	. 2 |- ( ( F Fn ( A X. B ) /\ G Fn ( C X. D ) ) -> ( F = G <-> ( ( A X. B ) = ( C X. D ) /\ A. z e. ( A X. B ) ( F ` z ) = ( G ` z ) ) ) )
2		fveq2 5414	. . . . . 6 |- ( z = <. x , y >. -> ( F ` z ) = ( F ` <. x , y >. ) )
3		fveq2 5414	. . . . . 6 |- ( z = <. x , y >. -> ( G ` z ) = ( G ` <. x , y >. ) )
4	2, 3	eqeq12d 2152	. . . . 5 |- ( z = <. x , y >. -> ( ( F ` z ) = ( G ` z ) <-> ( F ` <. x , y >. ) = ( G ` <. x , y >. ) ) )
5		df-ov 5770	. . . . . 6 |- ( x F y ) = ( F ` <. x , y >. )
6		df-ov 5770	. . . . . 6 |- ( x G y ) = ( G ` <. x , y >. )
7	5, 6	eqeq12i 2151	. . . . 5 |- ( ( x F y ) = ( x G y ) <-> ( F ` <. x , y >. ) = ( G ` <. x , y >. ) )
8	4, 7	syl6bbr 197	. . . 4 |- ( z = <. x , y >. -> ( ( F ` z ) = ( G ` z ) <-> ( x F y ) = ( x G y ) ) )
9	8	ralxp 4677	. . 3 |- ( A. z e. ( A X. B ) ( F ` z ) = ( G ` z ) <-> A. x e. A A. y e. B ( x F y ) = ( x G y ) )
10	9	anbi2i 452	. 2 |- ( ( ( A X. B ) = ( C X. D ) /\ A. z e. ( A X. B ) ( F ` z ) = ( G ` z ) ) <-> ( ( A X. B ) = ( C X. D ) /\ A. x e. A A. y e. B ( x F y ) = ( x G y ) ) )
11	1, 10	syl6bb 195	1 |- ( ( F Fn ( A X. B ) /\ G Fn ( C X. D ) ) -> ( F = G <-> ( ( A X. B ) = ( C X. D ) /\ A. x e. A A. y e. B ( x F y ) = ( x G y ) ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1331   A.wral 2414   <.cop 3525   X. cxp 4532   Fn wfn 5113   ` cfv 5118  (class class class)co 5767

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-sbc 2905  df-csb 2999  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-iota 5083  df-fun 5120  df-fn 5121  df-fv 5126  df-ov 5770

This theorem is referenced by:  eqfnov2  5871