Theorem elovimad 6057

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 6016	. 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 4756	. . 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 3226	. . . 4 |- ( ph -> <. A , B >. e. dom F )
8		funfvima 5881	. . . 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 2316	1 |- ( ph -> ( A F B ) e. ( F " ( C X. D ) ) )

Syntax hints:   -> wi 4   e. wcel 2200   C_ wss 3198   <.cop 3670   X. cxp 4721   dom cdm 4723   "cima 4726   Fun wfun 5318   ` cfv 5324  (class class class)co 6013

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2802  df-sbc 3030  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-br 4087  df-opab 4149  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-fv 5332  df-ov 6016

This theorem is referenced by: (None)