elhomai2 - Metamath Proof Explorer

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Theorem elhomai2 17749

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 17748	. . 3 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩(𝑋𝐻𝑌)𝐹)
10		df-br 5075	. . 3 ⊢ (⟨𝑋, 𝑌⟩(𝑋𝐻𝑌)𝐹 ↔ ⟨⟨𝑋, 𝑌⟩, 𝐹⟩ ∈ (𝑋𝐻𝑌))
11	9, 10	sylib 217	. 2 ⊢ (𝜑 → ⟨⟨𝑋, 𝑌⟩, 𝐹⟩ ∈ (𝑋𝐻𝑌))
12	1, 11	eqeltrid 2843	1 ⊢ (𝜑 → ⟨𝑋, 𝑌, 𝐹⟩ ∈ (𝑋𝐻𝑌))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2106 ⟨cop 4567 ⟨cotp 4569 class class class wbr 5074 ‘cfv 6433 (class class class)co 7275 Basecbs 16912 Hom chom 16973 Catccat 17373 Hom_achoma 17738

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-ot 4570 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-ov 7278 df-homa 17741

This theorem is referenced by: idahom 17775 coahom 17785

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