elhomai2 - Metamath Proof Explorer

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Theorem elhomai2 17288

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
elhomai2	⊢ (𝜑 → ⟨𝑋, 𝑌, 𝐹⟩ ∈ (𝑋𝐻𝑌))

**Proof of Theorem elhomai2**
Step	Hyp	Ref	Expression
1		df-ot 4570	. 2 ⊢ ⟨𝑋, 𝑌, 𝐹⟩ = ⟨⟨𝑋, 𝑌⟩, 𝐹⟩
2		homarcl.h	. . . 4 ⊢ 𝐻 = (Hom_a‘𝐶)
3		homafval.b	. . . 4 ⊢ 𝐵 = (Base‘𝐶)
4		homafval.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat)
5		homaval.j	. . . 4 ⊢ 𝐽 = (Hom ‘𝐶)
6		homaval.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
7		homaval.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
8		elhomai.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐽𝑌))
9	2, 3, 4, 5, 6, 7, 8	elhomai 17287	. . 3 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩(𝑋𝐻𝑌)𝐹)
10		df-br 5060	. . 3 ⊢ (⟨𝑋, 𝑌⟩(𝑋𝐻𝑌)𝐹 ↔ ⟨⟨𝑋, 𝑌⟩, 𝐹⟩ ∈ (𝑋𝐻𝑌))
11	9, 10	sylib 220	. 2 ⊢ (𝜑 → ⟨⟨𝑋, 𝑌⟩, 𝐹⟩ ∈ (𝑋𝐻𝑌))
12	1, 11	eqeltrid 2917	1 ⊢ (𝜑 → ⟨𝑋, 𝑌, 𝐹⟩ ∈ (𝑋𝐻𝑌))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 ⟨cop 4567 ⟨cotp 4569 class class class wbr 5059 ‘cfv 6350 (class class class)co 7150 Basecbs 16477 Hom chom 16570 Catccat 16929 Hom_achoma 17277

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-op 4568 df-ot 4570 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-ov 7153 df-homa 17280

This theorem is referenced by: idahom 17314 coahom 17324

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