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Theorem homadmcd 18004

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 17998	. . . 4 ⊢ Rel (𝑋𝐻𝑌)
3		1st2nd 8018	. . . 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 8021	. . . . . 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 17999	. . . . 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 4843	. . 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 4598	. 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 4595 ⟨cotp 4597 class class class wbr 5107 Rel wrel 5643 ‘cfv 6511 (class class class)co 7387 1^st c1st 7966 2^nd c2nd 7967 Hom_achoma 17985

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 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711

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 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-ot 4598 df-uni 4872 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-id 5533 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-ov 7390 df-1st 7968 df-2nd 7969 df-homa 17988

This theorem is referenced by: arwdmcd 18014 arwlid 18034 arwrid 18035

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