hof1 - Metamath Proof Explorer

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Theorem hof1 18243

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 18242	. . 3 ⊢ (𝜑 → (1^st ‘𝑀) = (Hom_f ‘𝐶))
4	3	oveqd 7431	. 2 ⊢ (𝜑 → (𝑋(1^st ‘𝑀)𝑌) = (𝑋(Hom_f ‘𝐶)𝑌))
5		eqid 2725	. . 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 17669	. 2 ⊢ (𝜑 → (𝑋(Hom_f ‘𝐶)𝑌) = (𝑋𝐻𝑌))
11	4, 10	eqtrd 2765	1 ⊢ (𝜑 → (𝑋(1^st ‘𝑀)𝑌) = (𝑋𝐻𝑌))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ‘cfv 6541 (class class class)co 7414 1^st c1st 7987 Basecbs 17177 Hom chom 17241 Catccat 17641 Hom_f chomf 17643 Hom_Fchof 18237

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5357 ax-pr 5421 ax-un 7736

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-nul 4317 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-iun 4991 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5568 df-xp 5676 df-rel 5677 df-cnv 5678 df-co 5679 df-dm 5680 df-rn 5681 df-res 5682 df-ima 5683 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-ov 7417 df-oprab 7418 df-mpo 7419 df-1st 7989 df-2nd 7990 df-homf 17647 df-hof 18239

This theorem is referenced by: yon11 18253 yonedalem21 18262

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