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Theorem homadmcd 17967

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 17961	. . . 4 ⊢ Rel (𝑋𝐻𝑌)
3		1st2nd 7981	. . . 4 ⊢ ((Rel (𝑋𝐻𝑌) ∧ 𝐹 ∈ (𝑋𝐻𝑌)) → 𝐹 = ⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩)
4	2, 3	mpan 690	. . 3 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 𝐹 = ⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩)
5		1st2ndbr 7984	. . . . . 6 ⊢ ((Rel (𝑋𝐻𝑌) ∧ 𝐹 ∈ (𝑋𝐻𝑌)) → (1^st ‘𝐹)(𝑋𝐻𝑌)(2^nd ‘𝐹))
6	2, 5	mpan 690	. . . . 5 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (1^st ‘𝐹)(𝑋𝐻𝑌)(2^nd ‘𝐹))
7	1	homa1 17962	. . . . 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 4833	. . 3 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → ⟨(1^st ‘𝐹), (2^nd ‘𝐹)⟩ = ⟨⟨𝑋, 𝑌⟩, (2^nd ‘𝐹)⟩)
10	4, 9	eqtrd 2764	. 2 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 𝐹 = ⟨⟨𝑋, 𝑌⟩, (2^nd ‘𝐹)⟩)
11		df-ot 4588	. 2 ⊢ ⟨𝑋, 𝑌, (2^nd ‘𝐹)⟩ = ⟨⟨𝑋, 𝑌⟩, (2^nd ‘𝐹)⟩
12	10, 11	eqtr4di 2782	1 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → 𝐹 = ⟨𝑋, 𝑌, (2^nd ‘𝐹)⟩)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2109 ⟨cop 4585 ⟨cotp 4587 class class class wbr 5095 Rel wrel 5628 ‘cfv 6486 (class class class)co 7353 1^st c1st 7929 2^nd c2nd 7930 Hom_achoma 17948

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7675

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-ral 3045 df-rex 3054 df-reu 3346 df-rab 3397 df-v 3440 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-nul 4287 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-op 4586 df-ot 4588 df-uni 4862 df-iun 4946 df-br 5096 df-opab 5158 df-mpt 5177 df-id 5518 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-ov 7356 df-1st 7931 df-2nd 7932 df-homa 17951

This theorem is referenced by: arwdmcd 17977 arwlid 17997 arwrid 17998

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