Theorem ovelimab 5671

Description: Operation value in an image. (Contributed by Mario Carneiro, 23-Dec-2013.) (Revised by Mario Carneiro, 29-Jan-2014.)

Assertion

Ref	Expression
ovelimab	|- ( ( F Fn A /\ ( B X. C ) C_ A ) -> ( D e. ( F " ( B X. C ) ) <-> E. x e. B E. y e. C D = ( x F y ) ) )

Distinct variable groups:   x, y, A   x, B, y   x, C, y   x, D, y   x, F, y

Proof of Theorem ovelimab

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvelimab 5250	. 2 |- ( ( F Fn A /\ ( B X. C ) C_ A ) -> ( D e. ( F " ( B X. C ) ) <-> E. z e. ( B X. C ) ( F ` z ) = D ) )
2		fveq2 5198	. . . . . 6 |- ( z = <. x , y >. -> ( F ` z ) = ( F ` <. x , y >. ) )
3		df-ov 5535	. . . . . 6 |- ( x F y ) = ( F ` <. x , y >. )
4	2, 3	syl6eqr 2131	. . . . 5 |- ( z = <. x , y >. -> ( F ` z ) = ( x F y ) )
5	4	eqeq1d 2089	. . . 4 |- ( z = <. x , y >. -> ( ( F ` z ) = D <-> ( x F y ) = D ) )
6		eqcom 2083	. . . 4 |- ( ( x F y ) = D <-> D = ( x F y ) )
7	5, 6	syl6bb 194	. . 3 |- ( z = <. x , y >. -> ( ( F ` z ) = D <-> D = ( x F y ) ) )
8	7	rexxp 4498	. 2 |- ( E. z e. ( B X. C ) ( F ` z ) = D <-> E. x e. B E. y e. C D = ( x F y ) )
9	1, 8	syl6bb 194	1 |- ( ( F Fn A /\ ( B X. C ) C_ A ) -> ( D e. ( F " ( B X. C ) ) <-> E. x e. B E. y e. C D = ( x F y ) ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1284   e. wcel 1433   E.wrex 2349   C_ wss 2973   <.cop 3401   X. cxp 4361   "cima 4366   Fn wfn 4917   ` cfv 4922  (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-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-fv 4930  df-ov 5535

This theorem is referenced by:  dfz2  8420  elq  8707