Theorem elovimad 6693

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 6653	. 2 |- ( A F B ) = ( F ` <. A , B >. )
2		elovimad.1	. . . 4 |- ( ph -> A e. C )
3		elovimad.2	. . . 4 |- ( ph -> B e. D )
4		opelxpi 5148	. . . 4 |- ( ( A e. C /\ B e. D ) -> <. A , B >. e. ( C X. D ) )
5	2, 3, 4	syl2anc 693	. . 3 |- ( ph -> <. A , B >. e. ( C X. D ) )
6		elovimad.3	. . . 4 |- ( ph -> Fun F )
7		elovimad.4	. . . . 5 |- ( ph -> ( C X. D ) C_ dom F )
8	7, 5	sseldd 3604	. . . 4 |- ( ph -> <. A , B >. e. dom F )
9		funfvima 6492	. . . 4 |- ( ( Fun F /\ <. A , B >. e. dom F ) -> ( <. A , B >. e. ( C X. D ) -> ( F ` <. A , B >. ) e. ( F " ( C X. D ) ) ) )
10	6, 8, 9	syl2anc 693	. . 3 |- ( ph -> ( <. A , B >. e. ( C X. D ) -> ( F ` <. A , B >. ) e. ( F " ( C X. D ) ) ) )
11	5, 10	mpd 15	. 2 |- ( ph -> ( F ` <. A , B >. ) e. ( F " ( C X. D ) ) )
12	1, 11	syl5eqel 2705	1 |- ( ph -> ( A F B ) e. ( F " ( C X. D ) ) )

Syntax hints:   -> wi 4   e. wcel 1990   C_ wss 3574   <.cop 4183   X. cxp 5112   dom cdm 5114   "cima 5117   Fun wfun 5882   ` cfv 5888  (class class class)co 6650

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896  df-ov 6653

This theorem is referenced by:  ltgov  25492  xrofsup  29533  icoreelrn  33209