Theorem hof2val 18213

Description: The morphism part of the Hom functor, for morphisms ⟨𝑓, 𝑔⟩:⟨𝑋, 𝑌⟩⟶⟨𝑍, 𝑊⟩ (which since the first argument is contravariant means morphisms 𝑓:𝑍⟶𝑋 and 𝑔:𝑌⟶𝑊), yields a function (a morphism of SetCat) mapping ℎ:𝑋⟶𝑌 to 𝑔 ∘ ℎ ∘ 𝑓:𝑍⟶𝑊. (Contributed by Mario Carneiro, 15-Jan-2017.)

Hypotheses

Ref	Expression
hofval.m	⊢ 𝑀 = (HomF‘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)
hof2.z	⊢ (𝜑 → 𝑍 ∈ 𝐵)
hof2.w	⊢ (𝜑 → 𝑊 ∈ 𝐵)
hof2.o	⊢ · = (comp‘𝐶)
hof2.f	⊢ (𝜑 → 𝐹 ∈ (𝑍𝐻𝑋))
hof2.g	⊢ (𝜑 → 𝐺 ∈ (𝑌𝐻𝑊))

Assertion

Ref	Expression
hof2val	⊢ (𝜑 → (𝐹(⟨𝑋, 𝑌⟩(2nd ‘𝑀)⟨𝑍, 𝑊⟩)𝐺) = (ℎ ∈ (𝑋𝐻𝑌) ↦ ((𝐺(⟨𝑋, 𝑌⟩ · 𝑊)ℎ)(⟨𝑍, 𝑋⟩ · 𝑊)𝐹)))

Distinct variable groups:   𝐵,ℎ   ℎ,𝐹   ℎ,𝐺   𝜑,ℎ   𝐶,ℎ   ℎ,𝐻   ℎ,𝑊   · ,ℎ   ℎ,𝑋   ℎ,𝑌   ℎ,𝑍

Allowed substitution hint:   𝑀(ℎ)

Proof of Theorem hof2val

Dummy variables 𝑓 𝑔 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		hof1.b	. . 3 ⊢ 𝐵 = (Base‘𝐶)
4		hof1.h	. . 3 ⊢ 𝐻 = (Hom ‘𝐶)
5		hof1.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
6		hof1.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
7		hof2.z	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
8		hof2.w	. . 3 ⊢ (𝜑 → 𝑊 ∈ 𝐵)
9		hof2.o	. . 3 ⊢ · = (comp‘𝐶)
10	1, 2, 3, 4, 5, 6, 7, 8, 9	hof2fval 18212	. 2 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩(2nd ‘𝑀)⟨𝑍, 𝑊⟩) = (𝑓 ∈ (𝑍𝐻𝑋), 𝑔 ∈ (𝑌𝐻𝑊) ↦ (ℎ ∈ (𝑋𝐻𝑌) ↦ ((𝑔(⟨𝑋, 𝑌⟩ · 𝑊)ℎ)(⟨𝑍, 𝑋⟩ · 𝑊)𝑓))))
11		simplrr 783	. . . . 5 ⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) ∧ ℎ ∈ (𝑋𝐻𝑌)) → 𝑔 = 𝐺)
12	11	oveq1d 7371	. . . 4 ⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) ∧ ℎ ∈ (𝑋𝐻𝑌)) → (𝑔(⟨𝑋, 𝑌⟩ · 𝑊)ℎ) = (𝐺(⟨𝑋, 𝑌⟩ · 𝑊)ℎ))
13		simplrl 782	. . . 4 ⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) ∧ ℎ ∈ (𝑋𝐻𝑌)) → 𝑓 = 𝐹)
14	12, 13	oveq12d 7374	. . 3 ⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) ∧ ℎ ∈ (𝑋𝐻𝑌)) → ((𝑔(⟨𝑋, 𝑌⟩ · 𝑊)ℎ)(⟨𝑍, 𝑋⟩ · 𝑊)𝑓) = ((𝐺(⟨𝑋, 𝑌⟩ · 𝑊)ℎ)(⟨𝑍, 𝑋⟩ · 𝑊)𝐹))
15	14	mpteq2dva 5165	. 2 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) → (ℎ ∈ (𝑋𝐻𝑌) ↦ ((𝑔(⟨𝑋, 𝑌⟩ · 𝑊)ℎ)(⟨𝑍, 𝑋⟩ · 𝑊)𝑓)) = (ℎ ∈ (𝑋𝐻𝑌) ↦ ((𝐺(⟨𝑋, 𝑌⟩ · 𝑊)ℎ)(⟨𝑍, 𝑋⟩ · 𝑊)𝐹)))
16		hof2.f	. 2 ⊢ (𝜑 → 𝐹 ∈ (𝑍𝐻𝑋))
17		hof2.g	. 2 ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐻𝑊))
18		ovex 7389	. . . 4 ⊢ (𝑋𝐻𝑌) ∈ V
19	18	mptex 7167	. . 3 ⊢ (ℎ ∈ (𝑋𝐻𝑌) ↦ ((𝐺(⟨𝑋, 𝑌⟩ · 𝑊)ℎ)(⟨𝑍, 𝑋⟩ · 𝑊)𝐹)) ∈ V
20	19	a1i 11	. 2 ⊢ (𝜑 → (ℎ ∈ (𝑋𝐻𝑌) ↦ ((𝐺(⟨𝑋, 𝑌⟩ · 𝑊)ℎ)(⟨𝑍, 𝑋⟩ · 𝑊)𝐹)) ∈ V)
21	10, 15, 16, 17, 20	ovmpod 7508	1 ⊢ (𝜑 → (𝐹(⟨𝑋, 𝑌⟩(2nd ‘𝑀)⟨𝑍, 𝑊⟩)𝐺) = (ℎ ∈ (𝑋𝐻𝑌) ↦ ((𝐺(⟨𝑋, 𝑌⟩ · 𝑊)ℎ)(⟨𝑍, 𝑋⟩ · 𝑊)𝐹)))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119  Vcvv 3431  ⟨cop 4561   ↦ cmpt 5153  ‘cfv 6485  (class class class)co 7356  2^nd c2nd 7930  Basecbs 17170  Hom chom 17222  compcco 17223  Catccat 17621  Hom_Fchof 18205

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-ov 7359  df-oprab 7360  df-mpo 7361  df-1st 7931  df-2nd 7932  df-hof 18207

This theorem is referenced by:  hof2  18214  hofcllem  18215  hofcl  18216  yonedalem3b  18236