Theorem funimassov 6026

Description: Membership relation for the values of a function whose image is a subclass. (Contributed by Mario Carneiro, 23-Dec-2013.)

Assertion

Ref	Expression
funimassov	|- ( ( Fun F /\ ( A X. B ) C_ dom F ) -> ( ( F " ( A X. B ) ) C_ C <-> A. x e. A A. y e. B ( x F y ) e. C ) )

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

Proof of Theorem funimassov

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		funimass4 5568	. 2 |- ( ( Fun F /\ ( A X. B ) C_ dom F ) -> ( ( F " ( A X. B ) ) C_ C <-> A. z e. ( A X. B ) ( F ` z ) e. C ) )
2		fveq2 5517	. . . . 5 |- ( z = <. x , y >. -> ( F ` z ) = ( F ` <. x , y >. ) )
3		df-ov 5880	. . . . 5 |- ( x F y ) = ( F ` <. x , y >. )
4	2, 3	eqtr4di 2228	. . . 4 |- ( z = <. x , y >. -> ( F ` z ) = ( x F y ) )
5	4	eleq1d 2246	. . 3 |- ( z = <. x , y >. -> ( ( F ` z ) e. C <-> ( x F y ) e. C ) )
6	5	ralxp 4772	. 2 |- ( A. z e. ( A X. B ) ( F ` z ) e. C <-> A. x e. A A. y e. B ( x F y ) e. C )
7	1, 6	bitrdi 196	1 |- ( ( Fun F /\ ( A X. B ) C_ dom F ) -> ( ( F " ( A X. B ) ) C_ C <-> A. x e. A A. y e. B ( x F y ) e. C ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1353   e. wcel 2148   A.wral 2455   C_ wss 3131   <.cop 3597   X. cxp 4626   dom cdm 4628   "cima 4631   Fun wfun 5212   ` cfv 5218  (class class class)co 5877

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-sbc 2965  df-csb 3060  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-iun 3890  df-br 4006  df-opab 4067  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-fv 5226  df-ov 5880

This theorem is referenced by: (None)