Theorem hof2fval 18143

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)

Assertion

Ref	Expression
hof2fval	⊢ (𝜑 → (⟨𝑋, 𝑌⟩(2nd ‘𝑀)⟨𝑍, 𝑊⟩) = (𝑓 ∈ (𝑍𝐻𝑋), 𝑔 ∈ (𝑌𝐻𝑊) ↦ (ℎ ∈ (𝑋𝐻𝑌) ↦ ((𝑔(⟨𝑋, 𝑌⟩ · 𝑊)ℎ)(⟨𝑍, 𝑋⟩ · 𝑊)𝑓))))

Distinct variable groups:   𝑓,𝑔,ℎ,𝐵   𝜑,𝑓,𝑔,ℎ   𝐶,𝑓,𝑔,ℎ   𝑓,𝐻,𝑔,ℎ   𝑓,𝑊,𝑔,ℎ   · ,𝑓,𝑔,ℎ   𝑓,𝑋,𝑔,ℎ   𝑓,𝑌,𝑔,ℎ   𝑓,𝑍,𝑔,ℎ

Allowed substitution hints:   𝑀(𝑓,𝑔,ℎ)

Proof of Theorem hof2fval

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

Step	Hyp	Ref	Expression
1		hofval.m	. . . 4 ⊢ 𝑀 = (HomF‘𝐶)
2		hofval.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat)
3		hof1.b	. . . 4 ⊢ 𝐵 = (Base‘𝐶)
4		hof1.h	. . . 4 ⊢ 𝐻 = (Hom ‘𝐶)
5		hof2.o	. . . 4 ⊢ · = (comp‘𝐶)
6	1, 2, 3, 4, 5	hofval 18140	. . 3 ⊢ (𝜑 → 𝑀 = ⟨(Homf ‘𝐶), (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ (𝐵 × 𝐵) ↦ (𝑓 ∈ ((1st ‘𝑦)𝐻(1st ‘𝑥)), 𝑔 ∈ ((2nd ‘𝑥)𝐻(2nd ‘𝑦)) ↦ (ℎ ∈ (𝐻‘𝑥) ↦ ((𝑔(𝑥 · (2nd ‘𝑦))ℎ)(⟨(1st ‘𝑦), (1st ‘𝑥)⟩ · (2nd ‘𝑦))𝑓))))⟩)
7		fvex 6855	. . . 4 ⊢ (Homf ‘𝐶) ∈ V
8	3	fvexi 6856	. . . . . 6 ⊢ 𝐵 ∈ V
9	8, 8	xpex 7686	. . . . 5 ⊢ (𝐵 × 𝐵) ∈ V
10	9, 9	mpoex 8011	. . . 4 ⊢ (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ (𝐵 × 𝐵) ↦ (𝑓 ∈ ((1st ‘𝑦)𝐻(1st ‘𝑥)), 𝑔 ∈ ((2nd ‘𝑥)𝐻(2nd ‘𝑦)) ↦ (ℎ ∈ (𝐻‘𝑥) ↦ ((𝑔(𝑥 · (2nd ‘𝑦))ℎ)(⟨(1st ‘𝑦), (1st ‘𝑥)⟩ · (2nd ‘𝑦))𝑓)))) ∈ V
11	7, 10	op2ndd 7931	. . 3 ⊢ (𝑀 = ⟨(Homf ‘𝐶), (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ (𝐵 × 𝐵) ↦ (𝑓 ∈ ((1st ‘𝑦)𝐻(1st ‘𝑥)), 𝑔 ∈ ((2nd ‘𝑥)𝐻(2nd ‘𝑦)) ↦ (ℎ ∈ (𝐻‘𝑥) ↦ ((𝑔(𝑥 · (2nd ‘𝑦))ℎ)(⟨(1st ‘𝑦), (1st ‘𝑥)⟩ · (2nd ‘𝑦))𝑓))))⟩ → (2nd ‘𝑀) = (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ (𝐵 × 𝐵) ↦ (𝑓 ∈ ((1st ‘𝑦)𝐻(1st ‘𝑥)), 𝑔 ∈ ((2nd ‘𝑥)𝐻(2nd ‘𝑦)) ↦ (ℎ ∈ (𝐻‘𝑥) ↦ ((𝑔(𝑥 · (2nd ‘𝑦))ℎ)(⟨(1st ‘𝑦), (1st ‘𝑥)⟩ · (2nd ‘𝑦))𝑓)))))
12	6, 11	syl 17	. 2 ⊢ (𝜑 → (2nd ‘𝑀) = (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ (𝐵 × 𝐵) ↦ (𝑓 ∈ ((1st ‘𝑦)𝐻(1st ‘𝑥)), 𝑔 ∈ ((2nd ‘𝑥)𝐻(2nd ‘𝑦)) ↦ (ℎ ∈ (𝐻‘𝑥) ↦ ((𝑔(𝑥 · (2nd ‘𝑦))ℎ)(⟨(1st ‘𝑦), (1st ‘𝑥)⟩ · (2nd ‘𝑦))𝑓)))))
13		simprr 771	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → 𝑦 = ⟨𝑍, 𝑊⟩)
14	13	fveq2d 6846	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → (1st ‘𝑦) = (1st ‘⟨𝑍, 𝑊⟩))
15		hof2.z	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
16		hof2.w	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ 𝐵)
17		op1stg 7932	. . . . . . 7 ⊢ ((𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → (1st ‘⟨𝑍, 𝑊⟩) = 𝑍)
18	15, 16, 17	syl2anc 584	. . . . . 6 ⊢ (𝜑 → (1st ‘⟨𝑍, 𝑊⟩) = 𝑍)
19	18	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → (1st ‘⟨𝑍, 𝑊⟩) = 𝑍)
20	14, 19	eqtrd 2776	. . . 4 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → (1st ‘𝑦) = 𝑍)
21		simprl 769	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → 𝑥 = ⟨𝑋, 𝑌⟩)
22	21	fveq2d 6846	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → (1st ‘𝑥) = (1st ‘⟨𝑋, 𝑌⟩))
23		hof1.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
24		hof1.y	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
25		op1stg 7932	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
26	23, 24, 25	syl2anc 584	. . . . . 6 ⊢ (𝜑 → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
27	26	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → (1st ‘⟨𝑋, 𝑌⟩) = 𝑋)
28	22, 27	eqtrd 2776	. . . 4 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → (1st ‘𝑥) = 𝑋)
29	20, 28	oveq12d 7374	. . 3 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → ((1st ‘𝑦)𝐻(1st ‘𝑥)) = (𝑍𝐻𝑋))
30	21	fveq2d 6846	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → (2nd ‘𝑥) = (2nd ‘⟨𝑋, 𝑌⟩))
31		op2ndg 7933	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
32	23, 24, 31	syl2anc 584	. . . . . 6 ⊢ (𝜑 → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
33	32	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → (2nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
34	30, 33	eqtrd 2776	. . . 4 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → (2nd ‘𝑥) = 𝑌)
35	13	fveq2d 6846	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → (2nd ‘𝑦) = (2nd ‘⟨𝑍, 𝑊⟩))
36		op2ndg 7933	. . . . . . 7 ⊢ ((𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → (2nd ‘⟨𝑍, 𝑊⟩) = 𝑊)
37	15, 16, 36	syl2anc 584	. . . . . 6 ⊢ (𝜑 → (2nd ‘⟨𝑍, 𝑊⟩) = 𝑊)
38	37	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → (2nd ‘⟨𝑍, 𝑊⟩) = 𝑊)
39	35, 38	eqtrd 2776	. . . 4 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → (2nd ‘𝑦) = 𝑊)
40	34, 39	oveq12d 7374	. . 3 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → ((2nd ‘𝑥)𝐻(2nd ‘𝑦)) = (𝑌𝐻𝑊))
41	21	fveq2d 6846	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → (𝐻‘𝑥) = (𝐻‘⟨𝑋, 𝑌⟩))
42		df-ov 7359	. . . . 5 ⊢ (𝑋𝐻𝑌) = (𝐻‘⟨𝑋, 𝑌⟩)
43	41, 42	eqtr4di 2794	. . . 4 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → (𝐻‘𝑥) = (𝑋𝐻𝑌))
44	20, 28	opeq12d 4838	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → ⟨(1st ‘𝑦), (1st ‘𝑥)⟩ = ⟨𝑍, 𝑋⟩)
45	44, 39	oveq12d 7374	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → (⟨(1st ‘𝑦), (1st ‘𝑥)⟩ · (2nd ‘𝑦)) = (⟨𝑍, 𝑋⟩ · 𝑊))
46	21, 39	oveq12d 7374	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → (𝑥 · (2nd ‘𝑦)) = (⟨𝑋, 𝑌⟩ · 𝑊))
47	46	oveqd 7373	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → (𝑔(𝑥 · (2nd ‘𝑦))ℎ) = (𝑔(⟨𝑋, 𝑌⟩ · 𝑊)ℎ))
48		eqidd 2737	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → 𝑓 = 𝑓)
49	45, 47, 48	oveq123d 7377	. . . 4 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → ((𝑔(𝑥 · (2nd ‘𝑦))ℎ)(⟨(1st ‘𝑦), (1st ‘𝑥)⟩ · (2nd ‘𝑦))𝑓) = ((𝑔(⟨𝑋, 𝑌⟩ · 𝑊)ℎ)(⟨𝑍, 𝑋⟩ · 𝑊)𝑓))
50	43, 49	mpteq12dv 5196	. . 3 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → (ℎ ∈ (𝐻‘𝑥) ↦ ((𝑔(𝑥 · (2nd ‘𝑦))ℎ)(⟨(1st ‘𝑦), (1st ‘𝑥)⟩ · (2nd ‘𝑦))𝑓)) = (ℎ ∈ (𝑋𝐻𝑌) ↦ ((𝑔(⟨𝑋, 𝑌⟩ · 𝑊)ℎ)(⟨𝑍, 𝑋⟩ · 𝑊)𝑓)))
51	29, 40, 50	mpoeq123dv 7431	. 2 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → (𝑓 ∈ ((1st ‘𝑦)𝐻(1st ‘𝑥)), 𝑔 ∈ ((2nd ‘𝑥)𝐻(2nd ‘𝑦)) ↦ (ℎ ∈ (𝐻‘𝑥) ↦ ((𝑔(𝑥 · (2nd ‘𝑦))ℎ)(⟨(1st ‘𝑦), (1st ‘𝑥)⟩ · (2nd ‘𝑦))𝑓))) = (𝑓 ∈ (𝑍𝐻𝑋), 𝑔 ∈ (𝑌𝐻𝑊) ↦ (ℎ ∈ (𝑋𝐻𝑌) ↦ ((𝑔(⟨𝑋, 𝑌⟩ · 𝑊)ℎ)(⟨𝑍, 𝑋⟩ · 𝑊)𝑓))))
52	23, 24	opelxpd 5671	. 2 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
53	15, 16	opelxpd 5671	. 2 ⊢ (𝜑 → ⟨𝑍, 𝑊⟩ ∈ (𝐵 × 𝐵))
54		ovex 7389	. . . 4 ⊢ (𝑍𝐻𝑋) ∈ V
55		ovex 7389	. . . 4 ⊢ (𝑌𝐻𝑊) ∈ V
56	54, 55	mpoex 8011	. . 3 ⊢ (𝑓 ∈ (𝑍𝐻𝑋), 𝑔 ∈ (𝑌𝐻𝑊) ↦ (ℎ ∈ (𝑋𝐻𝑌) ↦ ((𝑔(⟨𝑋, 𝑌⟩ · 𝑊)ℎ)(⟨𝑍, 𝑋⟩ · 𝑊)𝑓))) ∈ V
57	56	a1i 11	. 2 ⊢ (𝜑 → (𝑓 ∈ (𝑍𝐻𝑋), 𝑔 ∈ (𝑌𝐻𝑊) ↦ (ℎ ∈ (𝑋𝐻𝑌) ↦ ((𝑔(⟨𝑋, 𝑌⟩ · 𝑊)ℎ)(⟨𝑍, 𝑋⟩ · 𝑊)𝑓))) ∈ V)
58	12, 51, 52, 53, 57	ovmpod 7506	1 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩(2nd ‘𝑀)⟨𝑍, 𝑊⟩) = (𝑓 ∈ (𝑍𝐻𝑋), 𝑔 ∈ (𝑌𝐻𝑊) ↦ (ℎ ∈ (𝑋𝐻𝑌) ↦ ((𝑔(⟨𝑋, 𝑌⟩ · 𝑊)ℎ)(⟨𝑍, 𝑋⟩ · 𝑊)𝑓))))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  Vcvv 3445  ⟨cop 4592   ↦ cmpt 5188   × cxp 5631  ‘cfv 6496  (class class class)co 7356   ∈ cmpo 7358  1^st c1st 7918  2^nd c2nd 7919  Basecbs 17082  Hom chom 17143  compcco 17144  Catccat 17543  Hom_f chomf 17545  Hom_Fchof 18136

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7671

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7359  df-oprab 7360  df-mpo 7361  df-1st 7920  df-2nd 7921  df-hof 18138

This theorem is referenced by:  hof2val  18144