Theorem elovimad 6094

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 6053	. 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 4782	. . 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 3239	. . . 4 |- ( ph -> <. A , B >. e. dom F )
8		funfvima 5918	. . . 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 2319	1 |- ( ph -> ( A F B ) e. ( F " ( C X. D ) ) )

Syntax hints:   -> wi 4   e. wcel 2203   C_ wss 3211   <.cop 3692   X. cxp 4747   dom cdm 4749   "cima 4752   Fun wfun 5346   ` cfv 5352  (class class class)co 6050

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-sbc 3043  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-fv 5360  df-ov 6053

This theorem is referenced by: (None)