Theorem eqfnov2 5628

Description: Two operators with the same domain are equal iff their values at each point in the domain are equal. (Contributed by Jeff Madsen, 7-Jun-2010.)

Assertion

Ref	Expression
eqfnov2	|- ( ( F Fn ( A X. B ) /\ G Fn ( A X. B ) ) -> ( F = G <-> A. x e. A A. y e. B ( x F y ) = ( x G y ) ) )

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

Proof of Theorem eqfnov2

Step	Hyp	Ref	Expression
1		eqfnov 5627	. 2 |- ( ( F Fn ( A X. B ) /\ G Fn ( A X. B ) ) -> ( F = G <-> ( ( A X. B ) = ( A X. B ) /\ A. x e. A A. y e. B ( x F y ) = ( x G y ) ) ) )
2		simpr 108	. . 3 |- ( ( ( A X. B ) = ( A X. B ) /\ A. x e. A A. y e. B ( x F y ) = ( x G y ) ) -> A. x e. A A. y e. B ( x F y ) = ( x G y ) )
3		eqidd 2082	. . . 4 |- ( A. x e. A A. y e. B ( x F y ) = ( x G y ) -> ( A X. B ) = ( A X. B ) )
4	3	ancri 317	. . 3 |- ( A. x e. A A. y e. B ( x F y ) = ( x G y ) -> ( ( A X. B ) = ( A X. B ) /\ A. x e. A A. y e. B ( x F y ) = ( x G y ) ) )
5	2, 4	impbii 124	. 2 |- ( ( ( A X. B ) = ( A X. B ) /\ A. x e. A A. y e. B ( x F y ) = ( x G y ) ) <-> A. x e. A A. y e. B ( x F y ) = ( x G y ) )
6	1, 5	syl6bb 194	1 |- ( ( F Fn ( A X. B ) /\ G Fn ( A X. B ) ) -> ( F = G <-> A. x e. A A. y e. B ( x F y ) = ( x G y ) ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1284   A.wral 2348   X. cxp 4361   Fn wfn 4917  (class class class)co 5532

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-sbc 2816  df-csb 2909  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-iota 4887  df-fun 4924  df-fn 4925  df-fv 4930  df-ov 5535

This theorem is referenced by:  tpossym  5914