elhomai - Metamath Proof Explorer

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Theorem elhomai 17962

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 2738	. 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 17961	. 2 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩(𝑋𝐻𝑌)𝐹 ↔ (⟨𝑋, 𝑌⟩ = ⟨𝑋, 𝑌⟩ ∧ 𝐹 ∈ (𝑋𝐽𝑌))))
10	1, 2, 9	mpbir2and 714	1 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩(𝑋𝐻𝑌)𝐹)

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 ⟨cop 4587 class class class wbr 5099 ‘cfv 6493 (class class class)co 7361 Basecbs 17141 Hom chom 17193 Catccat 17592 Hom_achoma 17952

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5225 ax-sep 5242 ax-nul 5252 ax-pow 5311 ax-pr 5378 ax-un 7683

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3062 df-reu 3352 df-rab 3401 df-v 3443 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-nul 4287 df-if 4481 df-pw 4557 df-sn 4582 df-pr 4584 df-op 4588 df-uni 4865 df-iun 4949 df-br 5100 df-opab 5162 df-mpt 5181 df-id 5520 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-ov 7364 df-homa 17955

This theorem is referenced by: elhomai2 17963

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