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Theorem homadm 18221

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 18205	. . . 4 ⊢ dom_a = (1^st ∘ 1^st )
2	1	fveq1i 6887	. . 3 ⊢ (dom_a‘𝐹) = ((1^st ∘ 1^st )‘𝐹)
3		fo1st 8022	. . . . 5 ⊢ 1^st :V–onto→V
4		fof 6797	. . . . 5 ⊢ (1^st :V–onto→V → 1^st :V⟶V)
5	3, 4	ax-mp 5	. . . 4 ⊢ 1^st :V⟶V
6		elex 3471	. . . 4 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 𝐹 ∈ V)
7		fvco3 6986	. . . 4 ⊢ ((1^st :V⟶V ∧ 𝐹 ∈ V) → ((1^st ∘ 1^st )‘𝐹) = (1^st ‘(1^st ‘𝐹)))
8	5, 6, 7	sylancr 599	. . 3 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → ((1^st ∘ 1^st )‘𝐹) = (1^st ‘(1^st ‘𝐹)))
9	2, 8	eqtrid 2807	. 2 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (dom_a‘𝐹) = (1^st ‘(1^st ‘𝐹)))
10		homahom.h	. . . . . 6 ⊢ 𝐻 = (Hom_a‘𝐶)
11	10	homarel 18217	. . . . 5 ⊢ Rel (𝑋𝐻𝑌)
12		1st2ndbr 8054	. . . . 5 ⊢ ((Rel (𝑋𝐻𝑌) ∧ 𝐹 ∈ (𝑋𝐻𝑌)) → (1^st ‘𝐹)(𝑋𝐻𝑌)(2^nd ‘𝐹))
13	11, 12	mpan 703	. . . 4 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (1^st ‘𝐹)(𝑋𝐻𝑌)(2^nd ‘𝐹))
14	10	homa1 18218	. . . 4 ⊢ ((1^st ‘𝐹)(𝑋𝐻𝑌)(2^nd ‘𝐹) → (1^st ‘𝐹) = ⟨𝑋, 𝑌⟩)
15	13, 14	syl 18	. . 3 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (1^st ‘𝐹) = ⟨𝑋, 𝑌⟩)
16	15	fveq2d 6890	. 2 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (1^st ‘(1^st ‘𝐹)) = (1^st ‘⟨𝑋, 𝑌⟩))
17		eqid 2760	. . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶)
18	10, 17	homarcl2 18216	. . 3 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶)))
19		op1stg 8014	. . 3 ⊢ ((𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶)) → (1^st ‘⟨𝑋, 𝑌⟩) = 𝑋)
20	18, 19	syl 18	. 2 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (1^st ‘⟨𝑋, 𝑌⟩) = 𝑋)
21	9, 16, 20	3eqtrd 2799	1 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (dom_a‘𝐹) = 𝑋)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1569 ∈ wcel 2144 Vcvv 3450 ⟨cop 4834 class class class wbr 5102 ∘ ccom 5654 Rel wrel 5655 ⟶wf 6535 –onto→wfo 6537 ‘cfv 6539 (class class class)co 7421 1^st c1st 8000 2^nd c2nd 8001 Basecbs 17393 dom_acdoma 18201 Hom_achoma 18204

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7752

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-ral 3077 df-rex 3087 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-op 4835 df-uni 4867 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-id 5545 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-iota 6494 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-ov 7424 df-1st 8002 df-2nd 8003 df-doma 18205 df-homa 18207

This theorem is used by: arwhoma 18226 idadm 18242 homdmcoa 18248 coaval 18249

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