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Theorem homadm 16462

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 16446	. . . 4 ⊢ dom_a = (1^st ∘ 1^st )
2	1	fveq1i 6089	. . 3 ⊢ (dom_a‘𝐹) = ((1^st ∘ 1^st )‘𝐹)
3		fo1st 7057	. . . . 5 ⊢ 1^st :V–onto→V
4		fof 6013	. . . . 5 ⊢ (1^st :V–onto→V → 1^st :V⟶V)
5	3, 4	ax-mp 5	. . . 4 ⊢ 1^st :V⟶V
6		elex 3185	. . . 4 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 𝐹 ∈ V)
7		fvco3 6170	. . . 4 ⊢ ((1^st :V⟶V ∧ 𝐹 ∈ V) → ((1^st ∘ 1^st )‘𝐹) = (1^st ‘(1^st ‘𝐹)))
8	5, 6, 7	sylancr 694	. . 3 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → ((1^st ∘ 1^st )‘𝐹) = (1^st ‘(1^st ‘𝐹)))
9	2, 8	syl5eq 2656	. 2 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (dom_a‘𝐹) = (1^st ‘(1^st ‘𝐹)))
10		homahom.h	. . . . . 6 ⊢ 𝐻 = (Hom_a‘𝐶)
11	10	homarel 16458	. . . . 5 ⊢ Rel (𝑋𝐻𝑌)
12		1st2ndbr 7086	. . . . 5 ⊢ ((Rel (𝑋𝐻𝑌) ∧ 𝐹 ∈ (𝑋𝐻𝑌)) → (1^st ‘𝐹)(𝑋𝐻𝑌)(2^nd ‘𝐹))
13	11, 12	mpan 702	. . . 4 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (1^st ‘𝐹)(𝑋𝐻𝑌)(2^nd ‘𝐹))
14	10	homa1 16459	. . . 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 6092	. 2 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (1^st ‘(1^st ‘𝐹)) = (1^st ‘⟨𝑋, 𝑌⟩))
17		eqid 2610	. . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶)
18	10, 17	homarcl2 16457	. . 3 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶)))
19		op1stg 7049	. . 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 2648	1 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (dom_a‘𝐹) = 𝑋)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ⟨cop 4131 class class class wbr 4578 ∘ ccom 5032 Rel wrel 5033 ⟶wf 5786 –onto→wfo 5788 ‘cfv 5790 (class class class)co 6527 1^st c1st 7035 2^nd c2nd 7036 Basecbs 15644 dom_acdoma 16442 Hom_achoma 16445

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4694 ax-sep 4704 ax-nul 4712 ax-pow 4764 ax-pr 4828 ax-un 6825

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4368 df-iun 4452 df-br 4579 df-opab 4639 df-mpt 4640 df-id 4943 df-xp 5034 df-rel 5035 df-cnv 5036 df-co 5037 df-dm 5038 df-rn 5039 df-res 5040 df-ima 5041 df-iota 5754 df-fun 5792 df-fn 5793 df-f 5794 df-f1 5795 df-fo 5796 df-f1o 5797 df-fv 5798 df-ov 6530 df-1st 7037 df-2nd 7038 df-doma 16446 df-homa 16448

This theorem is referenced by: arwhoma 16467 idadm 16483 homdmcoa 16489 coaval 16490

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