Theorem eqfnov2 6055

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 6054	. 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 2206	. . . 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 1373   A.wral 2484   X. cxp 4674   Fn wfn 5267  (class class class)co 5946

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4163  ax-pow 4219  ax-pr 4254

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-sbc 2999  df-csb 3094  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-iun 3929  df-br 4046  df-opab 4107  df-mpt 4108  df-id 4341  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-iota 5233  df-fun 5274  df-fn 5275  df-fv 5280  df-ov 5949

This theorem is referenced by:  fnmpoovd  6303  tpossym  6364