Theorem ovelimab 6120

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 5658	. 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 5599	. . . . . 6 |- ( z = <. x , y >. -> ( F ` z ) = ( F ` <. x , y >. ) )
3		df-ov 5970	. . . . . 6 |- ( x F y ) = ( F ` <. x , y >. )
4	2, 3	eqtr4di 2258	. . . . 5 |- ( z = <. x , y >. -> ( F ` z ) = ( x F y ) )
5	4	eqeq1d 2216	. . . 4 |- ( z = <. x , y >. -> ( ( F ` z ) = D <-> ( x F y ) = D ) )
6		eqcom 2209	. . . 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 4840	. 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 1373   e. wcel 2178   E.wrex 2487   C_ wss 3174   <.cop 3646   X. cxp 4691   "cima 4696   Fn wfn 5285   ` cfv 5290  (class class class)co 5967

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-sbc 3006  df-csb 3102  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-iun 3943  df-br 4060  df-opab 4122  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-fv 5298  df-ov 5970

This theorem is referenced by:  dfz2  9480  elq  9778