elhomai - Metamath Proof Explorer

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Theorem elhomai 18044

Description: Produce an arrow from a morphism. (Contributed by Mario Carneiro, 11-Jan-2017.)

**Hypotheses**
Ref	Expression
homarcl.h	⊢ 𝐻 = (Hom_a‘𝐶)
homafval.b	⊢ 𝐵 = (Base‘𝐶)
homafval.c	⊢ (𝜑 → 𝐶 ∈ Cat)
homaval.j	⊢ 𝐽 = (Hom ‘𝐶)
homaval.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
homaval.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
elhomai.f	⊢ (𝜑 → 𝐹 ∈ (𝑋𝐽𝑌))

**Assertion**
Ref	Expression
elhomai	⊢ (𝜑 → ⟨𝑋, 𝑌⟩(𝑋𝐻𝑌)𝐹)

**Proof of Theorem elhomai**
Step	Hyp	Ref	Expression
1		eqidd 2736	. 2 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ = ⟨𝑋, 𝑌⟩)
2		elhomai.f	. 2 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐽𝑌))
3		homarcl.h	. . 3 ⊢ 𝐻 = (Hom_a‘𝐶)
4		homafval.b	. . 3 ⊢ 𝐵 = (Base‘𝐶)
5		homafval.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ Cat)
6		homaval.j	. . 3 ⊢ 𝐽 = (Hom ‘𝐶)
7		homaval.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
8		homaval.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
9	3, 4, 5, 6, 7, 8	elhoma 18043	. 2 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩(𝑋𝐻𝑌)𝐹 ↔ (⟨𝑋, 𝑌⟩ = ⟨𝑋, 𝑌⟩ ∧ 𝐹 ∈ (𝑋𝐽𝑌))))
10	1, 2, 9	mpbir2and 713	1 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩(𝑋𝐻𝑌)𝐹)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ⟨cop 4607 class class class wbr 5119 ‘cfv 6530 (class class class)co 7403 Basecbs 17226 Hom chom 17280 Catccat 17674 Hom_achoma 18034

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 7727

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 6483 df-fun 6532 df-fn 6533 df-f 6534 df-f1 6535 df-fo 6536 df-f1o 6537 df-fv 6538 df-ov 7406 df-homa 18037

This theorem is referenced by: elhomai2 18045

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