Theorem eqfnov2 6196

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 6195	. 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 110	. . 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 2239	. . . 4 |- ( A. x e. A A. y e. B ( x F y ) = ( x G y ) -> ( A X. B ) = ( A X. B ) )
4	3	ancri 324	. . 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 126	. 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	bitrdi 196	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 ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   A.wral 2528   X. cxp 4772   Fn wfn 5372  (class class class)co 6085

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-iota 5337  df-fun 5379  df-fn 5380  df-fv 5385  df-ov 6088

This theorem is used by:  fnmpoovd  6451  tpossym  6547