hof1 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > hof1		Structured version Visualization version GIF version

Theorem hof1 18454

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 18453	. . 3 ⊢ (𝜑 → (1^st ‘𝑀) = (Hom_f ‘𝐶))
4	3	oveqd 7457	. 2 ⊢ (𝜑 → (𝑋(1^st ‘𝑀)𝑌) = (𝑋(Hom_f ‘𝐶)𝑌))
5		eqid 2763	. . 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 17892	. 2 ⊢ (𝜑 → (𝑋(Hom_f ‘𝐶)𝑌) = (𝑋𝐻𝑌))
11	4, 10	eqtrd 2798	1 ⊢ (𝜑 → (𝑋(1^st ‘𝑀)𝑌) = (𝑋𝐻𝑌))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 = wceq 1570 ∈ wcel 2147 ‘cfv 6555 (class class class)co 7440 1^st c1st 8020 Basecbs 17413 Hom chom 17465 Catccat 17864 Hom_f chomf 17866 Hom_Fchof 18448

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5235 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7772

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-ral 3081 df-rex 3091 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-op 4839 df-uni 4871 df-iun 4956 df-br 5107 df-opab 5171 df-mpt 5190 df-id 5550 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-iota 6509 df-fun 6557 df-fn 6558 df-f 6559 df-f1 6560 df-fo 6561 df-f1o 6562 df-fv 6563 df-ov 7443 df-oprab 7444 df-mpo 7445 df-1st 8022 df-2nd 8023 df-homf 17870 df-hof 18450

This theorem is used by: yon11 18464 yonedalem21 18473

W3C validator