Theorem ovelimab 5929

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 5485	. 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 5429	. . . . . 6 |- ( z = <. x , y >. -> ( F ` z ) = ( F ` <. x , y >. ) )
3		df-ov 5785	. . . . . 6 |- ( x F y ) = ( F ` <. x , y >. )
4	2, 3	eqtr4di 2191	. . . . 5 |- ( z = <. x , y >. -> ( F ` z ) = ( x F y ) )
5	4	eqeq1d 2149	. . . 4 |- ( z = <. x , y >. -> ( ( F ` z ) = D <-> ( x F y ) = D ) )
6		eqcom 2142	. . . 4 |- ( ( x F y ) = D <-> D = ( x F y ) )
7	5, 6	syl6bb 195	. . 3 |- ( z = <. x , y >. -> ( ( F ` z ) = D <-> D = ( x F y ) ) )
8	7	rexxp 4691	. 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 195	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 103   <-> wb 104   = wceq 1332   e. wcel 1481   E.wrex 2418   C_ wss 3076   <.cop 3535   X. cxp 4545   "cima 4550   Fn wfn 5126   ` cfv 5131  (class class class)co 5782

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-sbc 2914  df-csb 3008  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-iun 3823  df-br 3938  df-opab 3998  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-fv 5139  df-ov 5785

This theorem is referenced by:  dfz2  9147  elq  9441