hof1 - Metamath Proof Explorer

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Theorem hof1 17487

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 17486	. . 3 ⊢ (𝜑 → (1^st ‘𝑀) = (Hom_f ‘𝐶))
4	3	oveqd 7159	. 2 ⊢ (𝜑 → (𝑋(1^st ‘𝑀)𝑌) = (𝑋(Hom_f ‘𝐶)𝑌))
5		eqid 2821	. . 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 16945	. 2 ⊢ (𝜑 → (𝑋(Hom_f ‘𝐶)𝑌) = (𝑋𝐻𝑌))
11	4, 10	eqtrd 2856	1 ⊢ (𝜑 → (𝑋(1^st ‘𝑀)𝑌) = (𝑋𝐻𝑌))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ‘cfv 6341 (class class class)co 7142 1^st c1st 7673 Basecbs 16466 Hom chom 16559 Catccat 16918 Hom_f chomf 16920 Hom_Fchof 17481

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5176 ax-sep 5189 ax-nul 5196 ax-pow 5252 ax-pr 5316 ax-un 7447

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3488 df-sbc 3764 df-csb 3872 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-nul 4280 df-if 4454 df-pw 4527 df-sn 4554 df-pr 4556 df-op 4560 df-uni 4825 df-iun 4907 df-br 5053 df-opab 5115 df-mpt 5133 df-id 5446 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-rn 5552 df-res 5553 df-ima 5554 df-iota 6300 df-fun 6343 df-fn 6344 df-f 6345 df-f1 6346 df-fo 6347 df-f1o 6348 df-fv 6349 df-ov 7145 df-oprab 7146 df-mpo 7147 df-1st 7675 df-2nd 7676 df-homf 16924 df-hof 17483

This theorem is referenced by: yon11 17497 yonedalem21 17506

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