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Theorem homadmcd 16739

Description: Decompose an arrow into domain, codomain, and morphism. (Contributed by Mario Carneiro, 11-Jan-2017.)

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

**Assertion**
Ref	Expression
homadmcd	⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 𝐹 = ⟨𝑋, 𝑌, (2^nd ‘𝐹)⟩)

**Proof of Theorem homadmcd**
Step	Hyp	Ref	Expression
1		homahom.h	. . . . 5 ⊢ 𝐻 = (Hom_a‘𝐶)
2	1	homarel 16733	. . . 4 ⊢ Rel (𝑋𝐻𝑌)
3		1st2nd 7258	. . . 4 ⊢ ((Rel (𝑋𝐻𝑌) ∧ 𝐹 ∈ (𝑋𝐻𝑌)) → 𝐹 = ⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩)
4	2, 3	mpan 706	. . 3 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 𝐹 = ⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩)
5		1st2ndbr 7261	. . . . . 6 ⊢ ((Rel (𝑋𝐻𝑌) ∧ 𝐹 ∈ (𝑋𝐻𝑌)) → (1^st ‘𝐹)(𝑋𝐻𝑌)(2^nd ‘𝐹))
6	2, 5	mpan 706	. . . . 5 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (1^st ‘𝐹)(𝑋𝐻𝑌)(2^nd ‘𝐹))
7	1	homa1 16734	. . . . 5 ⊢ ((1^st ‘𝐹)(𝑋𝐻𝑌)(2^nd ‘𝐹) → (1^st ‘𝐹) = ⟨𝑋, 𝑌⟩)
8	6, 7	syl 17	. . . 4 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (1^st ‘𝐹) = ⟨𝑋, 𝑌⟩)
9	8	opeq1d 4439	. . 3 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → ⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩ = ⟨⟨𝑋, 𝑌⟩, (2^nd ‘𝐹)⟩)
10	4, 9	eqtrd 2685	. 2 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 𝐹 = ⟨⟨𝑋, 𝑌⟩, (2^nd ‘𝐹)⟩)
11		df-ot 4219	. 2 ⊢ ⟨𝑋, 𝑌, (2^nd ‘𝐹)⟩ = ⟨⟨𝑋, 𝑌⟩, (2^nd ‘𝐹)⟩
12	10, 11	syl6eqr 2703	1 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 𝐹 = ⟨𝑋, 𝑌, (2^nd ‘𝐹)⟩)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1523 ∈ wcel 2030 ⟨cop 4216 ⟨cotp 4218 class class class wbr 4685 Rel wrel 5148 ‘cfv 5926 (class class class)co 6690 1^st c1st 7208 2^nd c2nd 7209 Hom_achoma 16720

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-op 4217 df-ot 4219 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-id 5053 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-ov 6693 df-1st 7210 df-2nd 7211 df-homa 16723

This theorem is referenced by: arwdmcd 16749 arwlid 16769 arwrid 16770

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