Theorem eqfnov 5959

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 5594	. 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 5496	. . . . . 6 |- ( z = <. x , y >. -> ( F ` z ) = ( F ` <. x , y >. ) )
3		fveq2 5496	. . . . . 6 |- ( z = <. x , y >. -> ( G ` z ) = ( G ` <. x , y >. ) )
4	2, 3	eqeq12d 2185	. . . . 5 |- ( z = <. x , y >. -> ( ( F ` z ) = ( G ` z ) <-> ( F ` <. x , y >. ) = ( G ` <. x , y >. ) ) )
5		df-ov 5856	. . . . . 6 |- ( x F y ) = ( F ` <. x , y >. )
6		df-ov 5856	. . . . . 6 |- ( x G y ) = ( G ` <. x , y >. )
7	5, 6	eqeq12i 2184	. . . . 5 |- ( ( x F y ) = ( x G y ) <-> ( F ` <. x , y >. ) = ( G ` <. x , y >. ) )
8	4, 7	bitr4di 197	. . . 4 |- ( z = <. x , y >. -> ( ( F ` z ) = ( G ` z ) <-> ( x F y ) = ( x G y ) ) )
9	8	ralxp 4754	. . 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 454	. 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	bitrdi 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 1348   A.wral 2448   <.cop 3586   X. cxp 4609   Fn wfn 5193   ` cfv 5198  (class class class)co 5853

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fn 5201  df-fv 5206  df-ov 5856

This theorem is referenced by:  eqfnov2  5960