Theorem hof2 18212

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))
hof2.k	⊢ (𝜑 → 𝐾 ∈ (𝑋𝐻𝑌))

Assertion

Ref	Expression
hof2	⊢ (𝜑 → ((𝐹(⟨𝑋, 𝑌⟩(2nd ‘𝑀)⟨𝑍, 𝑊⟩)𝐺)‘𝐾) = ((𝐺(⟨𝑋, 𝑌⟩ · 𝑊)𝐾)(⟨𝑍, 𝑋⟩ · 𝑊)𝐹))

Proof of Theorem hof2

Dummy variable ℎ is 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		hof2.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ (𝑍𝐻𝑋))
11		hof2.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐻𝑊))
12	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11	hof2val 18211	. 2 ⊢ (𝜑 → (𝐹(⟨𝑋, 𝑌⟩(2nd ‘𝑀)⟨𝑍, 𝑊⟩)𝐺) = (ℎ ∈ (𝑋𝐻𝑌) ↦ ((𝐺(⟨𝑋, 𝑌⟩ · 𝑊)ℎ)(⟨𝑍, 𝑋⟩ · 𝑊)𝐹)))
13		simpr 484	. . . 4 ⊢ ((𝜑 ∧ ℎ = 𝐾) → ℎ = 𝐾)
14	13	oveq2d 7372	. . 3 ⊢ ((𝜑 ∧ ℎ = 𝐾) → (𝐺(⟨𝑋, 𝑌⟩ · 𝑊)ℎ) = (𝐺(⟨𝑋, 𝑌⟩ · 𝑊)𝐾))
15	14	oveq1d 7371	. 2 ⊢ ((𝜑 ∧ ℎ = 𝐾) → ((𝐺(⟨𝑋, 𝑌⟩ · 𝑊)ℎ)(⟨𝑍, 𝑋⟩ · 𝑊)𝐹) = ((𝐺(⟨𝑋, 𝑌⟩ · 𝑊)𝐾)(⟨𝑍, 𝑋⟩ · 𝑊)𝐹))
16		hof2.k	. 2 ⊢ (𝜑 → 𝐾 ∈ (𝑋𝐻𝑌))
17		ovexd 7391	. 2 ⊢ (𝜑 → ((𝐺(⟨𝑋, 𝑌⟩ · 𝑊)𝐾)(⟨𝑍, 𝑋⟩ · 𝑊)𝐹) ∈ V)
18	12, 15, 16, 17	fvmptd 6944	1 ⊢ (𝜑 → ((𝐹(⟨𝑋, 𝑌⟩(2nd ‘𝑀)⟨𝑍, 𝑊⟩)𝐺)‘𝐾) = ((𝐺(⟨𝑋, 𝑌⟩ · 𝑊)𝐾)(⟨𝑍, 𝑋⟩ · 𝑊)𝐹))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Vcvv 3427  ⟨cop 4563  ‘cfv 6487  (class class class)co 7356  2^nd c2nd 7930  Basecbs 17168  Hom chom 17220  compcco 17221  Catccat 17619  Hom_Fchof 18203

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 2184  ax-ext 2707  ax-rep 5201  ax-sep 5220  ax-nul 5230  ax-pow 5296  ax-pr 5364  ax-un 7678

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 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2931  df-ral 3050  df-rex 3060  df-reu 3341  df-rab 3388  df-v 3429  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4264  df-if 4457  df-pw 4533  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-iun 4925  df-br 5075  df-opab 5137  df-mpt 5156  df-id 5515  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-f1 6492  df-fo 6493  df-f1o 6494  df-fv 6495  df-ov 7359  df-oprab 7360  df-mpo 7361  df-1st 7931  df-2nd 7932  df-hof 18205

This theorem is referenced by:  yon12  18220  yon2  18221