Theorem elovimad 6119

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 6078	. 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 4802	. . 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 3249	. . . 4 |- ( ph -> <. A , B >. e. dom F )
8		funfvima 5940	. . . 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 415	. . 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 2325	1 |- ( ph -> ( A F B ) e. ( F " ( C X. D ) ) )

Syntax hints:   -> wi 4   e. wcel 2209   C_ wss 3220   <.cop 3708   X. cxp 4767   dom cdm 4769   "cima 4772   Fun wfun 5366   ` cfv 5372  (class class class)co 6075

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-fv 5380  df-ov 6078

This theorem is referenced by: (None)