Theorem eqfnov 6766

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 6312	. 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 6191	. . . . . 6 |- ( z = <. x , y >. -> ( F ` z ) = ( F ` <. x , y >. ) )
3		fveq2 6191	. . . . . 6 |- ( z = <. x , y >. -> ( G ` z ) = ( G ` <. x , y >. ) )
4	2, 3	eqeq12d 2637	. . . . 5 |- ( z = <. x , y >. -> ( ( F ` z ) = ( G ` z ) <-> ( F ` <. x , y >. ) = ( G ` <. x , y >. ) ) )
5		df-ov 6653	. . . . . 6 |- ( x F y ) = ( F ` <. x , y >. )
6		df-ov 6653	. . . . . 6 |- ( x G y ) = ( G ` <. x , y >. )
7	5, 6	eqeq12i 2636	. . . . 5 |- ( ( x F y ) = ( x G y ) <-> ( F ` <. x , y >. ) = ( G ` <. x , y >. ) )
8	4, 7	syl6bbr 278	. . . 4 |- ( z = <. x , y >. -> ( ( F ` z ) = ( G ` z ) <-> ( x F y ) = ( x G y ) ) )
9	8	ralxp 5263	. . 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 730	. 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 276	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   <-> wb 196   /\ wa 384   = wceq 1483   A.wral 2912   <.cop 4183   X. cxp 5112   Fn wfn 5883   ` cfv 5888  (class class class)co 6650

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896  df-ov 6653

This theorem is referenced by:  eqfnov2  6767  oprres  6802  ssceq  16486  sspg  27583  ssps  27585  sspmlem  27587