elhomai - Metamath Proof Explorer

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Theorem elhomai 18000

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 17999	. 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 4574 class class class wbr 5086 ‘cfv 6499 (class class class)co 7367 Basecbs 17179 Hom chom 17231 Catccat 17630 Hom_achoma 17990

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 5213 ax-sep 5232 ax-nul 5242 ax-pow 5308 ax-pr 5376 ax-un 7689

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 3063 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-id 5526 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6455 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-ov 7370 df-homa 17993

This theorem is referenced by: elhomai2 18001

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