Theorem hof1fval 18312

Description: The object part of the Hom functor is the Hom_f operation, which is just a functionalized version of Hom. That is, it is a two argument function, which 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 = (HomF‘𝐶)
hofval.c	⊢ (𝜑 → 𝐶 ∈ Cat)

Assertion

Ref	Expression
hof1fval	⊢ (𝜑 → (1st ‘𝑀) = (Homf ‘𝐶))

Proof of Theorem hof1fval

Dummy variables 𝑓 𝑔 ℎ 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		hofval.m	. . 3 ⊢ 𝑀 = (HomF‘𝐶)
2		hofval.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ Cat)
3		eqid 2770	. . 3 ⊢ (Base‘𝐶) = (Base‘𝐶)
4		eqid 2770	. . 3 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
5		eqid 2770	. . 3 ⊢ (comp‘𝐶) = (comp‘𝐶)
6	1, 2, 3, 4, 5	hofval 18311	. 2 ⊢ (𝜑 → 𝑀 = ⟨(Homf ‘𝐶), (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ↦ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)), 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)) ↦ (ℎ ∈ ((Hom ‘𝐶)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝐶)(2nd ‘𝑦))ℎ)(⟨(1st ‘𝑦), (1st ‘𝑥)⟩(comp‘𝐶)(2nd ‘𝑦))𝑓))))⟩)
7		fvex 6898	. . 3 ⊢ (Homf ‘𝐶) ∈ V
8		fvex 6898	. . . . 5 ⊢ (Base‘𝐶) ∈ V
9	8, 8	xpex 7755	. . . 4 ⊢ ((Base‘𝐶) × (Base‘𝐶)) ∈ V
10	9, 9	mpoex 8079	. . 3 ⊢ (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ↦ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)), 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)) ↦ (ℎ ∈ ((Hom ‘𝐶)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝐶)(2nd ‘𝑦))ℎ)(⟨(1st ‘𝑦), (1st ‘𝑥)⟩(comp‘𝐶)(2nd ‘𝑦))𝑓)))) ∈ V
11	7, 10	op1std 7999	. 2 ⊢ (𝑀 = ⟨(Homf ‘𝐶), (𝑥 ∈ ((Base‘𝐶) × (Base‘𝐶)), 𝑦 ∈ ((Base‘𝐶) × (Base‘𝐶)) ↦ (𝑓 ∈ ((1st ‘𝑦)(Hom ‘𝐶)(1st ‘𝑥)), 𝑔 ∈ ((2nd ‘𝑥)(Hom ‘𝐶)(2nd ‘𝑦)) ↦ (ℎ ∈ ((Hom ‘𝐶)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝐶)(2nd ‘𝑦))ℎ)(⟨(1st ‘𝑦), (1st ‘𝑥)⟩(comp‘𝐶)(2nd ‘𝑦))𝑓))))⟩ → (1st ‘𝑀) = (Homf ‘𝐶))
12	6, 11	syl 18	1 ⊢ (𝜑 → (1st ‘𝑀) = (Homf ‘𝐶))

Syntax hints:   → wi 4   = wceq 1568   ∈ wcel 2150  ⟨cop 4600   ↦ cmpt 5197   × cxp 5663  ‘cfv 6540  (class class class)co 7414   ∈ cmpo 7416  1^st c1st 7987  2^nd c2nd 7988  Basecbs 17272  Hom chom 17324  compcco 17325  Catccat 17723  Hom_f chomf 17725  Hom_Fchof 18307

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-10 2183  ax-11 2199  ax-12 2220  ax-ext 2742  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pow 5340  ax-pr 5408  ax-un 7736

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2099  df-mo 2574  df-eu 2604  df-clab 2749  df-cleq 2762  df-clel 2845  df-nfc 2919  df-ne 2966  df-ral 3087  df-rex 3097  df-reu 3377  df-rab 3424  df-v 3464  df-sbc 3753  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5560  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7417  df-oprab 7418  df-mpo 7419  df-1st 7989  df-2nd 7990  df-hof 18309

This theorem is referenced by:  hof1  18313