Theorem hof2 17891

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 17890	. 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 7271	. . 3 ⊢ ((𝜑 ∧ ℎ = 𝐾) → (𝐺(⟨𝑋, 𝑌⟩ · 𝑊)ℎ) = (𝐺(⟨𝑋, 𝑌⟩ · 𝑊)𝐾))
15	14	oveq1d 7270	. 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 7290	. 2 ⊢ (𝜑 → ((𝐺(⟨𝑋, 𝑌⟩ · 𝑊)𝐾)(⟨𝑍, 𝑋⟩ · 𝑊)𝐹) ∈ V)
18	12, 15, 16, 17	fvmptd 6864	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 1539   ∈ wcel 2108  Vcvv 3422  ⟨cop 4564  ‘cfv 6418  (class class class)co 7255  2^nd c2nd 7803  Basecbs 16840  Hom chom 16899  compcco 16900  Catccat 17290  Hom_Fchof 17882

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-1st 7804  df-2nd 7805  df-hof 17884

This theorem is referenced by:  yon12  17899  yon2  17900