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Theorem homadm 17998

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 17982	. . . 4 ⊢ dom_a = (1^st ∘ 1^st )
2	1	fveq1i 6835	. . 3 ⊢ (dom_a‘𝐹) = ((1^st ∘ 1^st )‘𝐹)
3		fo1st 7955	. . . . 5 ⊢ 1^st :V–onto→V
4		fof 6746	. . . . 5 ⊢ (1^st :V–onto→V → 1^st :V⟶V)
5	3, 4	ax-mp 5	. . . 4 ⊢ 1^st :V⟶V
6		elex 3451	. . . 4 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 𝐹 ∈ V)
7		fvco3 6933	. . . 4 ⊢ ((1^st :V⟶V ∧ 𝐹 ∈ V) → ((1^st ∘ 1^st )‘𝐹) = (1^st ‘(1^st ‘𝐹)))
8	5, 6, 7	sylancr 588	. . 3 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → ((1^st ∘ 1^st )‘𝐹) = (1^st ‘(1^st ‘𝐹)))
9	2, 8	eqtrid 2784	. 2 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (dom_a‘𝐹) = (1^st ‘(1^st ‘𝐹)))
10		homahom.h	. . . . . 6 ⊢ 𝐻 = (Hom_a‘𝐶)
11	10	homarel 17994	. . . . 5 ⊢ Rel (𝑋𝐻𝑌)
12		1st2ndbr 7988	. . . . 5 ⊢ ((Rel (𝑋𝐻𝑌) ∧ 𝐹 ∈ (𝑋𝐻𝑌)) → (1^st ‘𝐹)(𝑋𝐻𝑌)(2^nd ‘𝐹))
13	11, 12	mpan 691	. . . 4 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (1^st ‘𝐹)(𝑋𝐻𝑌)(2^nd ‘𝐹))
14	10	homa1 17995	. . . 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 6838	. 2 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (1^st ‘(1^st ‘𝐹)) = (1^st ‘⟨𝑋, 𝑌⟩))
17		eqid 2737	. . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶)
18	10, 17	homarcl2 17993	. . 3 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶)))
19		op1stg 7947	. . 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 2776	1 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (dom_a‘𝐹) = 𝑋)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1542 ∈ wcel 2114 Vcvv 3430 ⟨cop 4574 class class class wbr 5086 ∘ ccom 5628 Rel wrel 5629 ⟶wf 6488 –onto→wfo 6490 ‘cfv 6492 (class class class)co 7360 1^st c1st 7933 2^nd c2nd 7934 Basecbs 17170 dom_acdoma 17978 Hom_achoma 17981

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5302 ax-pr 5370 ax-un 7682

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3063 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-id 5519 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-ov 7363 df-1st 7935 df-2nd 7936 df-doma 17982 df-homa 17984

This theorem is referenced by: arwhoma 18003 idadm 18019 homdmcoa 18025 coaval 18026

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