Theorem eqfnov 6004

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 5635	. 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 5534	. . . . . 6 |- ( z = <. x , y >. -> ( F ` z ) = ( F ` <. x , y >. ) )
3		fveq2 5534	. . . . . 6 |- ( z = <. x , y >. -> ( G ` z ) = ( G ` <. x , y >. ) )
4	2, 3	eqeq12d 2204	. . . . 5 |- ( z = <. x , y >. -> ( ( F ` z ) = ( G ` z ) <-> ( F ` <. x , y >. ) = ( G ` <. x , y >. ) ) )
5		df-ov 5900	. . . . . 6 |- ( x F y ) = ( F ` <. x , y >. )
6		df-ov 5900	. . . . . 6 |- ( x G y ) = ( G ` <. x , y >. )
7	5, 6	eqeq12i 2203	. . . . 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 4788	. . 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 1364   A.wral 2468   <.cop 3610   X. cxp 4642   Fn wfn 5230   ` cfv 5235  (class class class)co 5897

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-csb 3073  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-iota 5196  df-fun 5237  df-fn 5238  df-fv 5243  df-ov 5900

This theorem is referenced by:  eqfnov2  6005