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Theorem homadm 18022

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 18006	. . . 4 ⊢ dom_a = (1^st ∘ 1^st )
2	1	fveq1i 6892	. . 3 ⊢ (dom_a‘𝐹) = ((1^st ∘ 1^st )‘𝐹)
3		fo1st 8007	. . . . 5 ⊢ 1^st :V–onto→V
4		fof 6805	. . . . 5 ⊢ (1^st :V–onto→V → 1^st :V⟶V)
5	3, 4	ax-mp 5	. . . 4 ⊢ 1^st :V⟶V
6		elex 3489	. . . 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 586	. . 3 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → ((1^st ∘ 1^st )‘𝐹) = (1^st ‘(1^st ‘𝐹)))
9	2, 8	eqtrid 2780	. 2 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (dom_a‘𝐹) = (1^st ‘(1^st ‘𝐹)))
10		homahom.h	. . . . . 6 ⊢ 𝐻 = (Hom_a‘𝐶)
11	10	homarel 18018	. . . . 5 ⊢ Rel (𝑋𝐻𝑌)
12		1st2ndbr 8040	. . . . 5 ⊢ ((Rel (𝑋𝐻𝑌) ∧ 𝐹 ∈ (𝑋𝐻𝑌)) → (1^st ‘𝐹)(𝑋𝐻𝑌)(2^nd ‘𝐹))
13	11, 12	mpan 689	. . . 4 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (1^st ‘𝐹)(𝑋𝐻𝑌)(2^nd ‘𝐹))
14	10	homa1 18019	. . . 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 6895	. 2 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (1^st ‘(1^st ‘𝐹)) = (1^st ‘⟨𝑋, 𝑌⟩))
17		eqid 2728	. . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶)
18	10, 17	homarcl2 18017	. . 3 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶)))
19		op1stg 7999	. . 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 2772	1 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (dom_a‘𝐹) = 𝑋)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 Vcvv 3470 ⟨cop 4630 class class class wbr 5142 ∘ ccom 5676 Rel wrel 5677 ⟶wf 6538 –onto→wfo 6540 ‘cfv 6542 (class class class)co 7414 1^st c1st 7985 2^nd c2nd 7986 Basecbs 17173 dom_acdoma 18002 Hom_achoma 18005

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7417 df-1st 7987 df-2nd 7988 df-doma 18006 df-homa 18008

This theorem is referenced by: arwhoma 18027 idadm 18043 homdmcoa 18049 coaval 18050

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