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Theorem homadm 17996

Description: The domain of an arrow with known domain and codomain. (Contributed by Mario Carneiro, 11-Jan-2017.)

**Hypothesis**
Ref	Expression
homahom.h	⊢ 𝐻 = (Hom_a‘𝐶)

**Assertion**
Ref	Expression
homadm	⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (dom_a‘𝐹) = 𝑋)

**Proof of Theorem homadm**
Step	Hyp	Ref	Expression
1		df-doma 17980	. . . 4 ⊢ dom_a = (1^st ∘ 1^st )
2	1	fveq1i 6893	. . 3 ⊢ (dom_a‘𝐹) = ((1^st ∘ 1^st )‘𝐹)
3		fo1st 7999	. . . . 5 ⊢ 1^st :V–onto→V
4		fof 6806	. . . . 5 ⊢ (1^st :V–onto→V → 1^st :V⟶V)
5	3, 4	ax-mp 5	. . . 4 ⊢ 1^st :V⟶V
6		elex 3491	. . . 4 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 𝐹 ∈ V)
7		fvco3 6991	. . . 4 ⊢ ((1^st :V⟶V ∧ 𝐹 ∈ V) → ((1^st ∘ 1^st )‘𝐹) = (1^st ‘(1^st ‘𝐹)))
8	5, 6, 7	sylancr 585	. . 3 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → ((1^st ∘ 1^st )‘𝐹) = (1^st ‘(1^st ‘𝐹)))
9	2, 8	eqtrid 2782	. 2 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (dom_a‘𝐹) = (1^st ‘(1^st ‘𝐹)))
10		homahom.h	. . . . . 6 ⊢ 𝐻 = (Hom_a‘𝐶)
11	10	homarel 17992	. . . . 5 ⊢ Rel (𝑋𝐻𝑌)
12		1st2ndbr 8032	. . . . 5 ⊢ ((Rel (𝑋𝐻𝑌) ∧ 𝐹 ∈ (𝑋𝐻𝑌)) → (1^st ‘𝐹)(𝑋𝐻𝑌)(2^nd ‘𝐹))
13	11, 12	mpan 686	. . . 4 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (1^st ‘𝐹)(𝑋𝐻𝑌)(2^nd ‘𝐹))
14	10	homa1 17993	. . . 4 ⊢ ((1^st ‘𝐹)(𝑋𝐻𝑌)(2^nd ‘𝐹) → (1^st ‘𝐹) = ⟨𝑋, 𝑌⟩)
15	13, 14	syl 17	. . 3 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (1^st ‘𝐹) = ⟨𝑋, 𝑌⟩)
16	15	fveq2d 6896	. 2 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (1^st ‘(1^st ‘𝐹)) = (1^st ‘⟨𝑋, 𝑌⟩))
17		eqid 2730	. . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶)
18	10, 17	homarcl2 17991	. . 3 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶)))
19		op1stg 7991	. . 3 ⊢ ((𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶)) → (1^st ‘⟨𝑋, 𝑌⟩) = 𝑋)
20	18, 19	syl 17	. 2 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (1^st ‘⟨𝑋, 𝑌⟩) = 𝑋)
21	9, 16, 20	3eqtrd 2774	1 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (dom_a‘𝐹) = 𝑋)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 = wceq 1539 ∈ wcel 2104 Vcvv 3472 ⟨cop 4635 class class class wbr 5149 ∘ ccom 5681 Rel wrel 5682 ⟶wf 6540 –onto→wfo 6542 ‘cfv 6544 (class class class)co 7413 1^st c1st 7977 2^nd c2nd 7978 Basecbs 17150 dom_acdoma 17976 Hom_achoma 17979

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7729

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-ov 7416 df-1st 7979 df-2nd 7980 df-doma 17980 df-homa 17982

This theorem is referenced by: arwhoma 18001 idadm 18017 homdmcoa 18023 coaval 18024

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