Theorem eqfnov 6138

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 5754	. 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 5648	. . . . . 6 |- ( z = <. x , y >. -> ( F ` z ) = ( F ` <. x , y >. ) )
3		fveq2 5648	. . . . . 6 |- ( z = <. x , y >. -> ( G ` z ) = ( G ` <. x , y >. ) )
4	2, 3	eqeq12d 2246	. . . . 5 |- ( z = <. x , y >. -> ( ( F ` z ) = ( G ` z ) <-> ( F ` <. x , y >. ) = ( G ` <. x , y >. ) ) )
5		df-ov 6031	. . . . . 6 |- ( x F y ) = ( F ` <. x , y >. )
6		df-ov 6031	. . . . . 6 |- ( x G y ) = ( G ` <. x , y >. )
7	5, 6	eqeq12i 2245	. . . . 5 |- ( ( x F y ) = ( x G y ) <-> ( F ` <. x , y >. ) = ( G ` <. x , y >. ) )
8	4, 7	bitr4di 198	. . . 4 |- ( z = <. x , y >. -> ( ( F ` z ) = ( G ` z ) <-> ( x F y ) = ( x G y ) ) )
9	8	ralxp 4879	. . 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 457	. 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 196	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 104   <-> wb 105   = wceq 1398   A.wral 2511   <.cop 3676   X. cxp 4729   Fn wfn 5328   ` cfv 5333  (class class class)co 6028

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-sbc 3033  df-csb 3129  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-iota 5293  df-fun 5335  df-fn 5336  df-fv 5341  df-ov 6031

This theorem is referenced by:  eqfnov2  6139