hof1 - Metamath Proof Explorer

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Theorem hof1 18000

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 17999	. . 3 ⊢ (𝜑 → (1^st ‘𝑀) = (Hom_f ‘𝐶))
4	3	oveqd 7312	. 2 ⊢ (𝜑 → (𝑋(1^st ‘𝑀)𝑌) = (𝑋(Hom_f ‘𝐶)𝑌))
5		eqid 2733	. . 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 17429	. 2 ⊢ (𝜑 → (𝑋(Hom_f ‘𝐶)𝑌) = (𝑋𝐻𝑌))
11	4, 10	eqtrd 2773	1 ⊢ (𝜑 → (𝑋(1^st ‘𝑀)𝑌) = (𝑋𝐻𝑌))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2101 ‘cfv 6447 (class class class)co 7295 1^st c1st 7849 Basecbs 16940 Hom chom 17001 Catccat 17401 Hom_f chomf 17403 Hom_Fchof 17994

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2103 ax-9 2111 ax-10 2132 ax-11 2149 ax-12 2166 ax-ext 2704 ax-rep 5212 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7608

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2063 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2884 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3223 df-rab 3224 df-v 3436 df-sbc 3719 df-csb 3835 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-nul 4260 df-if 4463 df-pw 4538 df-sn 4565 df-pr 4567 df-op 4571 df-uni 4842 df-iun 4929 df-br 5078 df-opab 5140 df-mpt 5161 df-id 5491 df-xp 5597 df-rel 5598 df-cnv 5599 df-co 5600 df-dm 5601 df-rn 5602 df-res 5603 df-ima 5604 df-iota 6399 df-fun 6449 df-fn 6450 df-f 6451 df-f1 6452 df-fo 6453 df-f1o 6454 df-fv 6455 df-ov 7298 df-oprab 7299 df-mpo 7300 df-1st 7851 df-2nd 7852 df-homf 17407 df-hof 17996

This theorem is referenced by: yon11 18010 yonedalem21 18019

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