Theorem eqfnov2 6030

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 6029	. 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 2197	. . . 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 ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1364   A.wral 2475   X. cxp 4661   Fn wfn 5253  (class class class)co 5922

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-csb 3085  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-iota 5219  df-fun 5260  df-fn 5261  df-fv 5266  df-ov 5925

This theorem is referenced by:  fnmpoovd  6273  tpossym  6334