Theorem ovelimab 6172

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 5702	. 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 5639	. . . . . 6 |- ( z = <. x , y >. -> ( F ` z ) = ( F ` <. x , y >. ) )
3		df-ov 6020	. . . . . 6 |- ( x F y ) = ( F ` <. x , y >. )
4	2, 3	eqtr4di 2282	. . . . 5 |- ( z = <. x , y >. -> ( F ` z ) = ( x F y ) )
5	4	eqeq1d 2240	. . . 4 |- ( z = <. x , y >. -> ( ( F ` z ) = D <-> ( x F y ) = D ) )
6		eqcom 2233	. . . 4 |- ( ( x F y ) = D <-> D = ( x F y ) )
7	5, 6	bitrdi 196	. . 3 |- ( z = <. x , y >. -> ( ( F ` z ) = D <-> D = ( x F y ) ) )
8	7	rexxp 4874	. 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	bitrdi 196	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 104   <-> wb 105   = wceq 1397   e. wcel 2202   E.wrex 2511   C_ wss 3200   <.cop 3672   X. cxp 4723   "cima 4728   Fn wfn 5321   ` cfv 5326  (class class class)co 6017

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-csb 3128  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-iun 3972  df-br 4089  df-opab 4151  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-fv 5334  df-ov 6020

This theorem is referenced by:  dfz2  9551  elq  9855