Theorem elovimad 6061

Description: Elementhood of the image set of an operation value. (Contributed by Thierry Arnoux, 13-Mar-2017.)

Hypotheses

Ref	Expression
elovimad.1	|- ( ph -> A e. C )
elovimad.2	|- ( ph -> B e. D )
elovimad.3	|- ( ph -> Fun F )
elovimad.4	|- ( ph -> ( C X. D ) C_ dom F )

Assertion

Ref	Expression
elovimad	|- ( ph -> ( A F B ) e. ( F " ( C X. D ) ) )

Proof of Theorem elovimad

Step	Hyp	Ref	Expression
1		df-ov 6020	. 2 |- ( A F B ) = ( F ` <. A , B >. )
2		elovimad.1	. . . 4 |- ( ph -> A e. C )
3		elovimad.2	. . . 4 |- ( ph -> B e. D )
4	2, 3	opelxpd 4758	. . 3 |- ( ph -> <. A , B >. e. ( C X. D ) )
5		elovimad.3	. . . 4 |- ( ph -> Fun F )
6		elovimad.4	. . . . 5 |- ( ph -> ( C X. D ) C_ dom F )
7	6, 4	sseldd 3228	. . . 4 |- ( ph -> <. A , B >. e. dom F )
8		funfvima 5885	. . . 4 |- ( ( Fun F /\ <. A , B >. e. dom F ) -> ( <. A , B >. e. ( C X. D ) -> ( F ` <. A , B >. ) e. ( F " ( C X. D ) ) ) )
9	5, 7, 8	syl2anc 411	. . 3 |- ( ph -> ( <. A , B >. e. ( C X. D ) -> ( F ` <. A , B >. ) e. ( F " ( C X. D ) ) ) )
10	4, 9	mpd 13	. 2 |- ( ph -> ( F ` <. A , B >. ) e. ( F " ( C X. D ) ) )
11	1, 10	eqeltrid 2318	1 |- ( ph -> ( A F B ) e. ( F " ( C X. D ) ) )

Syntax hints:   -> wi 4   e. wcel 2202   C_ wss 3200   <.cop 3672   X. cxp 4723   dom cdm 4725   "cima 4728   Fun wfun 5320   ` 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-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  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: (None)