hof1 - Metamath Proof Explorer

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Theorem hof1 18209

Description: The object part of the Hom functor maps 𝑋, 𝑌 to the set of morphisms from 𝑋 to 𝑌. (Contributed by Mario Carneiro, 15-Jan-2017.)

**Hypotheses**
Ref	Expression
hofval.m	⊢ 𝑀 = (Hom_F‘𝐶)
hofval.c	⊢ (𝜑 → 𝐶 ∈ Cat)
hof1.b	⊢ 𝐵 = (Base‘𝐶)
hof1.h	⊢ 𝐻 = (Hom ‘𝐶)
hof1.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
hof1.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)

**Assertion**
Ref	Expression
hof1	⊢ (𝜑 → (𝑋(1^st ‘𝑀)𝑌) = (𝑋𝐻𝑌))

**Proof of Theorem hof1**
Step	Hyp	Ref	Expression
1		hofval.m	. . . 4 ⊢ 𝑀 = (Hom_F‘𝐶)
2		hofval.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat)
3	1, 2	hof1fval 18208	. . 3 ⊢ (𝜑 → (1^st ‘𝑀) = (Hom_f ‘𝐶))
4	3	oveqd 7373	. 2 ⊢ (𝜑 → (𝑋(1^st ‘𝑀)𝑌) = (𝑋(Hom_f ‘𝐶)𝑌))
5		eqid 2735	. . 3 ⊢ (Hom_f ‘𝐶) = (Hom_f ‘𝐶)
6		hof1.b	. . 3 ⊢ 𝐵 = (Base‘𝐶)
7		hof1.h	. . 3 ⊢ 𝐻 = (Hom ‘𝐶)
8		hof1.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
9		hof1.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
10	5, 6, 7, 8, 9	homfval 17647	. 2 ⊢ (𝜑 → (𝑋(Hom_f ‘𝐶)𝑌) = (𝑋𝐻𝑌))
11	4, 10	eqtrd 2770	1 ⊢ (𝜑 → (𝑋(1^st ‘𝑀)𝑌) = (𝑋𝐻𝑌))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 ‘cfv 6487 (class class class)co 7356 1^st c1st 7929 Basecbs 17168 Hom chom 17220 Catccat 17619 Hom_f chomf 17621 Hom_Fchof 18203

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 2184 ax-ext 2707 ax-rep 5201 ax-sep 5220 ax-nul 5230 ax-pow 5296 ax-pr 5364 ax-un 7678

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 2538 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2810 df-nfc 2884 df-ne 2931 df-ral 3050 df-rex 3060 df-reu 3341 df-rab 3388 df-v 3429 df-sbc 3726 df-csb 3834 df-dif 3888 df-un 3890 df-in 3892 df-ss 3902 df-nul 4264 df-if 4457 df-pw 4533 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4841 df-iun 4925 df-br 5075 df-opab 5137 df-mpt 5156 df-id 5515 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-iota 6443 df-fun 6489 df-fn 6490 df-f 6491 df-f1 6492 df-fo 6493 df-f1o 6494 df-fv 6495 df-ov 7359 df-oprab 7360 df-mpo 7361 df-1st 7931 df-2nd 7932 df-homf 17625 df-hof 18205

This theorem is referenced by: yon11 18219 yonedalem21 18228

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