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Theorem homahom2 18051

Description: The second component of an arrow is the corresponding morphism (without the domain/codomain tag). (Contributed by Mario Carneiro, 11-Jan-2017.)

**Hypotheses**
Ref	Expression
homahom.h	⊢ 𝐻 = (Hom_a‘𝐶)
homahom.j	⊢ 𝐽 = (Hom ‘𝐶)

**Assertion**
Ref	Expression
homahom2	⊢ (𝑍(𝑋𝐻𝑌)𝐹 → 𝐹 ∈ (𝑋𝐽𝑌))

**Proof of Theorem homahom2**
Step	Hyp	Ref	Expression
1		df-br 5120	. . . 4 ⊢ (𝑍(𝑋𝐻𝑌)𝐹 ↔ ⟨𝑍, 𝐹⟩ ∈ (𝑋𝐻𝑌))
2		homahom.h	. . . . 5 ⊢ 𝐻 = (Hom_a‘𝐶)
3		eqid 2735	. . . . 5 ⊢ (Base‘𝐶) = (Base‘𝐶)
4	2	homarcl 18041	. . . . 5 ⊢ (⟨𝑍, 𝐹⟩ ∈ (𝑋𝐻𝑌) → 𝐶 ∈ Cat)
5		homahom.j	. . . . 5 ⊢ 𝐽 = (Hom ‘𝐶)
6	2, 3	homarcl2 18048	. . . . . 6 ⊢ (⟨𝑍, 𝐹⟩ ∈ (𝑋𝐻𝑌) → (𝑋 ∈ (Base‘𝐶) ∧ 𝑌 ∈ (Base‘𝐶)))
7	6	simpld 494	. . . . 5 ⊢ (⟨𝑍, 𝐹⟩ ∈ (𝑋𝐻𝑌) → 𝑋 ∈ (Base‘𝐶))
8	6	simprd 495	. . . . 5 ⊢ (⟨𝑍, 𝐹⟩ ∈ (𝑋𝐻𝑌) → 𝑌 ∈ (Base‘𝐶))
9	2, 3, 4, 5, 7, 8	elhoma 18045	. . . 4 ⊢ (⟨𝑍, 𝐹⟩ ∈ (𝑋𝐻𝑌) → (𝑍(𝑋𝐻𝑌)𝐹 ↔ (𝑍 = ⟨𝑋, 𝑌⟩ ∧ 𝐹 ∈ (𝑋𝐽𝑌))))
10	1, 9	sylbi 217	. . 3 ⊢ (𝑍(𝑋𝐻𝑌)𝐹 → (𝑍(𝑋𝐻𝑌)𝐹 ↔ (𝑍 = ⟨𝑋, 𝑌⟩ ∧ 𝐹 ∈ (𝑋𝐽𝑌))))
11	10	ibi 267	. 2 ⊢ (𝑍(𝑋𝐻𝑌)𝐹 → (𝑍 = ⟨𝑋, 𝑌⟩ ∧ 𝐹 ∈ (𝑋𝐽𝑌)))
12	11	simprd 495	1 ⊢ (𝑍(𝑋𝐻𝑌)𝐹 → 𝐹 ∈ (𝑋𝐽𝑌))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ⟨cop 4607 class class class wbr 5119 ‘cfv 6531 (class class class)co 7405 Basecbs 17228 Hom chom 17282 Hom_achoma 18036

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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7729

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 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-ral 3052 df-rex 3061 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-op 4608 df-uni 4884 df-iun 4969 df-br 5120 df-opab 5182 df-mpt 5202 df-id 5548 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-ov 7408 df-homa 18039

This theorem is referenced by: homahom 18052

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