Theorem eqfnov2 6112

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 6111	. 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 2230	. . . 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 1395   A.wral 2508   X. cxp 4717   Fn wfn 5313  (class class class)co 6001

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-csb 3125  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-iota 5278  df-fun 5320  df-fn 5321  df-fv 5326  df-ov 6004

This theorem is referenced by:  fnmpoovd  6361  tpossym  6422