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Theorem homadmcd 17978

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 17972	. . . 4 ⊢ Rel (𝑋𝐻𝑌)
3		1st2nd 7993	. . . 4 ⊢ ((Rel (𝑋𝐻𝑌) ∧ 𝐹 ∈ (𝑋𝐻𝑌)) → 𝐹 = ⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩)
4	2, 3	mpan 691	. . 3 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 𝐹 = ⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩)
5		1st2ndbr 7996	. . . . . 6 ⊢ ((Rel (𝑋𝐻𝑌) ∧ 𝐹 ∈ (𝑋𝐻𝑌)) → (1^st ‘𝐹)(𝑋𝐻𝑌)(2^nd ‘𝐹))
6	2, 5	mpan 691	. . . . 5 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (1^st ‘𝐹)(𝑋𝐻𝑌)(2^nd ‘𝐹))
7	1	homa1 17973	. . . . 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 4837	. . 3 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → ⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩ = ⟨⟨𝑋, 𝑌⟩, (2^nd ‘𝐹)⟩)
10	4, 9	eqtrd 2772	. 2 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 𝐹 = ⟨⟨𝑋, 𝑌⟩, (2^nd ‘𝐹)⟩)
11		df-ot 4591	. 2 ⊢ ⟨𝑋, 𝑌, (2^nd ‘𝐹)⟩ = ⟨⟨𝑋, 𝑌⟩, (2^nd ‘𝐹)⟩
12	10, 11	eqtr4di 2790	1 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 𝐹 = ⟨𝑋, 𝑌, (2^nd ‘𝐹)⟩)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 ⟨cop 4588 ⟨cotp 4590 class class class wbr 5100 Rel wrel 5637 ‘cfv 6500 (class class class)co 7368 1^st c1st 7941 2^nd c2nd 7942 Hom_achoma 17959

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 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690

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 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-ot 4591 df-uni 4866 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-id 5527 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-ov 7371 df-1st 7943 df-2nd 7944 df-homa 17962

This theorem is referenced by: arwdmcd 17988 arwlid 18008 arwrid 18009

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