homadmcd - Metamath Proof Explorer

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Theorem homadmcd 18249

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 18243	. . . 4 ⊢ Rel (𝑋𝐻𝑌)
3		1st2nd 8077	. . . 4 ⊢ ((Rel (𝑋𝐻𝑌) ∧ 𝐹 ∈ (𝑋𝐻𝑌)) → 𝐹 = ⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩)
4	2, 3	mpan 703	. . 3 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 𝐹 = ⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩)
5		1st2ndbr 8080	. . . . . 6 ⊢ ((Rel (𝑋𝐻𝑌) ∧ 𝐹 ∈ (𝑋𝐻𝑌)) → (1^st ‘𝐹)(𝑋𝐻𝑌)(2^nd ‘𝐹))
6	2, 5	mpan 703	. . . . 5 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (1^st ‘𝐹)(𝑋𝐻𝑌)(2^nd ‘𝐹))
7	1	homa1 18244	. . . . 5 ⊢ ((1^st ‘𝐹)(𝑋𝐻𝑌)(2^nd ‘𝐹) → (1^st ‘𝐹) = ⟨𝑋, 𝑌⟩)
8	6, 7	syl 18	. . . 4 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (1^st ‘𝐹) = ⟨𝑋, 𝑌⟩)
9	8	opeq1d 4849	. . 3 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → ⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩ = ⟨⟨𝑋, 𝑌⟩, (2^nd ‘𝐹)⟩)
10	4, 9	eqtrd 2798	. 2 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 𝐹 = ⟨⟨𝑋, 𝑌⟩, (2^nd ‘𝐹)⟩)
11		df-ot 4862	. 2 ⊢ ⟨𝑋, 𝑌, (2^nd ‘𝐹)⟩ = ⟨⟨𝑋, 𝑌⟩, (2^nd ‘𝐹)⟩
12	10, 11	eqtr4di 2816	1 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 𝐹 = ⟨𝑋, 𝑌, (2^nd ‘𝐹)⟩)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 = wceq 1570 ∈ wcel 2147 ⟨cop 4838 ⟨cotp 4861 class class class wbr 5106 Rel wrel 5661 ‘cfv 6561 (class class class)co 7446 1^st c1st 8026 2^nd c2nd 8027 Hom_achoma 18230

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5236 ax-sep 5253 ax-nul 5264 ax-pow 5332 ax-pr 5396 ax-un 7778

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-ral 3081 df-rex 3091 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4839 df-ot 4862 df-uni 4871 df-iun 4956 df-br 5107 df-opab 5172 df-mpt 5191 df-id 5551 df-xp 5662 df-rel 5663 df-cnv 5664 df-co 5665 df-dm 5666 df-rn 5667 df-res 5668 df-ima 5669 df-iota 6515 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7449 df-1st 8028 df-2nd 8029 df-homa 18233

This theorem is used by: arwdmcd 18259 arwlid 18279 arwrid 18280

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