Theorem hof2val 18191

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 18190	. 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 778	. . . . 5 ⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) ∧ ℎ ∈ (𝑋𝐻𝑌)) → 𝑔 = 𝐺)
12	11	oveq1d 7383	. . . 4 ⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) ∧ ℎ ∈ (𝑋𝐻𝑌)) → (𝑔(⟨𝑋, 𝑌⟩ · 𝑊)ℎ) = (𝐺(⟨𝑋, 𝑌⟩ · 𝑊)ℎ))
13		simplrl 777	. . . 4 ⊢ (((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) ∧ ℎ ∈ (𝑋𝐻𝑌)) → 𝑓 = 𝐹)
14	12, 13	oveq12d 7386	. . 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 5193	. 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 7401	. . . 4 ⊢ (𝑋𝐻𝑌) ∈ V
19	18	mptex 7179	. . 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 7520	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 395   = wceq 1542   ∈ wcel 2114  Vcvv 3442  ⟨cop 4588   ↦ cmpt 5181  ‘cfv 6500  (class class class)co 7368  2^nd c2nd 7942  Basecbs 17148  Hom chom 17200  compcco 17201  Catccat 17599  Hom_Fchof 18183

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-1st 7943  df-2nd 7944  df-hof 18185

This theorem is referenced by:  hof2  18192  hofcllem  18193  hofcl  18194  yonedalem3b  18214