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Theorem hof2fval 18220

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 = (Hom_F‘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⟩(2^nd ‘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 = (Hom_F‘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 18217	. . 3 ⊢ (𝜑 → 𝑀 = ⟨(Hom_f ‘𝐶), (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ (𝐵 × 𝐵) ↦ (𝑓 ∈ ((1^st ‘𝑦)𝐻(1^st ‘𝑥)), 𝑔 ∈ ((2^nd ‘𝑥)𝐻(2^nd ‘𝑦)) ↦ (ℎ ∈ (𝐻‘𝑥) ↦ ((𝑔(𝑥 · (2^nd ‘𝑦))ℎ)(⟨(1^st ‘𝑦), (1^st ‘𝑥)⟩ · (2^nd ‘𝑦))𝑓))))⟩)
7		fvex 6898	. . . 4 ⊢ (Hom_f ‘𝐶) ∈ V
8	3	fvexi 6899	. . . . . 6 ⊢ 𝐵 ∈ V
9	8, 8	xpex 7737	. . . . 5 ⊢ (𝐵 × 𝐵) ∈ V
10	9, 9	mpoex 8065	. . . 4 ⊢ (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ (𝐵 × 𝐵) ↦ (𝑓 ∈ ((1^st ‘𝑦)𝐻(1^st ‘𝑥)), 𝑔 ∈ ((2^nd ‘𝑥)𝐻(2^nd ‘𝑦)) ↦ (ℎ ∈ (𝐻‘𝑥) ↦ ((𝑔(𝑥 · (2^nd ‘𝑦))ℎ)(⟨(1^st ‘𝑦), (1^st ‘𝑥)⟩ · (2^nd ‘𝑦))𝑓)))) ∈ V
11	7, 10	op2ndd 7985	. . 3 ⊢ (𝑀 = ⟨(Hom_f ‘𝐶), (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ (𝐵 × 𝐵) ↦ (𝑓 ∈ ((1^st ‘𝑦)𝐻(1^st ‘𝑥)), 𝑔 ∈ ((2^nd ‘𝑥)𝐻(2^nd ‘𝑦)) ↦ (ℎ ∈ (𝐻‘𝑥) ↦ ((𝑔(𝑥 · (2^nd ‘𝑦))ℎ)(⟨(1^st ‘𝑦), (1^st ‘𝑥)⟩ · (2^nd ‘𝑦))𝑓))))⟩ → (2^nd ‘𝑀) = (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ (𝐵 × 𝐵) ↦ (𝑓 ∈ ((1^st ‘𝑦)𝐻(1^st ‘𝑥)), 𝑔 ∈ ((2^nd ‘𝑥)𝐻(2^nd ‘𝑦)) ↦ (ℎ ∈ (𝐻‘𝑥) ↦ ((𝑔(𝑥 · (2^nd ‘𝑦))ℎ)(⟨(1^st ‘𝑦), (1^st ‘𝑥)⟩ · (2^nd ‘𝑦))𝑓)))))
12	6, 11	syl 17	. 2 ⊢ (𝜑 → (2^nd ‘𝑀) = (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ (𝐵 × 𝐵) ↦ (𝑓 ∈ ((1^st ‘𝑦)𝐻(1^st ‘𝑥)), 𝑔 ∈ ((2^nd ‘𝑥)𝐻(2^nd ‘𝑦)) ↦ (ℎ ∈ (𝐻‘𝑥) ↦ ((𝑔(𝑥 · (2^nd ‘𝑦))ℎ)(⟨(1^st ‘𝑦), (1^st ‘𝑥)⟩ · (2^nd ‘𝑦))𝑓)))))
13		simprr 770	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → 𝑦 = ⟨𝑍, 𝑊⟩)
14	13	fveq2d 6889	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → (1^st ‘𝑦) = (1^st ‘⟨𝑍, 𝑊⟩))
15		hof2.z	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
16		hof2.w	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ 𝐵)
17		op1stg 7986	. . . . . . 7 ⊢ ((𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → (1^st ‘⟨𝑍, 𝑊⟩) = 𝑍)
18	15, 16, 17	syl2anc 583	. . . . . 6 ⊢ (𝜑 → (1^st ‘⟨𝑍, 𝑊⟩) = 𝑍)
19	18	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → (1^st ‘⟨𝑍, 𝑊⟩) = 𝑍)
20	14, 19	eqtrd 2766	. . . 4 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → (1^st ‘𝑦) = 𝑍)
21		simprl 768	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → 𝑥 = ⟨𝑋, 𝑌⟩)
22	21	fveq2d 6889	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → (1^st ‘𝑥) = (1^st ‘⟨𝑋, 𝑌⟩))
23		hof1.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
24		hof1.y	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
25		op1stg 7986	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (1^st ‘⟨𝑋, 𝑌⟩) = 𝑋)
26	23, 24, 25	syl2anc 583	. . . . . 6 ⊢ (𝜑 → (1^st ‘⟨𝑋, 𝑌⟩) = 𝑋)
27	26	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → (1^st ‘⟨𝑋, 𝑌⟩) = 𝑋)
28	22, 27	eqtrd 2766	. . . 4 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → (1^st ‘𝑥) = 𝑋)
29	20, 28	oveq12d 7423	. . 3 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → ((1^st ‘𝑦)𝐻(1^st ‘𝑥)) = (𝑍𝐻𝑋))
30	21	fveq2d 6889	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → (2^nd ‘𝑥) = (2^nd ‘⟨𝑋, 𝑌⟩))
31		op2ndg 7987	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (2^nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
32	23, 24, 31	syl2anc 583	. . . . . 6 ⊢ (𝜑 → (2^nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
33	32	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → (2^nd ‘⟨𝑋, 𝑌⟩) = 𝑌)
34	30, 33	eqtrd 2766	. . . 4 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → (2^nd ‘𝑥) = 𝑌)
35	13	fveq2d 6889	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → (2^nd ‘𝑦) = (2^nd ‘⟨𝑍, 𝑊⟩))
36		op2ndg 7987	. . . . . . 7 ⊢ ((𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → (2^nd ‘⟨𝑍, 𝑊⟩) = 𝑊)
37	15, 16, 36	syl2anc 583	. . . . . 6 ⊢ (𝜑 → (2^nd ‘⟨𝑍, 𝑊⟩) = 𝑊)
38	37	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → (2^nd ‘⟨𝑍, 𝑊⟩) = 𝑊)
39	35, 38	eqtrd 2766	. . . 4 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → (2^nd ‘𝑦) = 𝑊)
40	34, 39	oveq12d 7423	. . 3 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → ((2^nd ‘𝑥)𝐻(2^nd ‘𝑦)) = (𝑌𝐻𝑊))
41	21	fveq2d 6889	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → (𝐻‘𝑥) = (𝐻‘⟨𝑋, 𝑌⟩))
42		df-ov 7408	. . . . 5 ⊢ (𝑋𝐻𝑌) = (𝐻‘⟨𝑋, 𝑌⟩)
43	41, 42	eqtr4di 2784	. . . 4 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → (𝐻‘𝑥) = (𝑋𝐻𝑌))
44	20, 28	opeq12d 4876	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → ⟨(1^st ‘𝑦), (1^st ‘𝑥)⟩ = ⟨𝑍, 𝑋⟩)
45	44, 39	oveq12d 7423	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → (⟨(1^st ‘𝑦), (1^st ‘𝑥)⟩ · (2^nd ‘𝑦)) = (⟨𝑍, 𝑋⟩ · 𝑊))
46	21, 39	oveq12d 7423	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → (𝑥 · (2^nd ‘𝑦)) = (⟨𝑋, 𝑌⟩ · 𝑊))
47	46	oveqd 7422	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → (𝑔(𝑥 · (2^nd ‘𝑦))ℎ) = (𝑔(⟨𝑋, 𝑌⟩ · 𝑊)ℎ))
48		eqidd 2727	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → 𝑓 = 𝑓)
49	45, 47, 48	oveq123d 7426	. . . 4 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → ((𝑔(𝑥 · (2^nd ‘𝑦))ℎ)(⟨(1^st ‘𝑦), (1^st ‘𝑥)⟩ · (2^nd ‘𝑦))𝑓) = ((𝑔(⟨𝑋, 𝑌⟩ · 𝑊)ℎ)(⟨𝑍, 𝑋⟩ · 𝑊)𝑓))
50	43, 49	mpteq12dv 5232	. . 3 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → (ℎ ∈ (𝐻‘𝑥) ↦ ((𝑔(𝑥 · (2^nd ‘𝑦))ℎ)(⟨(1^st ‘𝑦), (1^st ‘𝑥)⟩ · (2^nd ‘𝑦))𝑓)) = (ℎ ∈ (𝑋𝐻𝑌) ↦ ((𝑔(⟨𝑋, 𝑌⟩ · 𝑊)ℎ)(⟨𝑍, 𝑋⟩ · 𝑊)𝑓)))
51	29, 40, 50	mpoeq123dv 7480	. 2 ⊢ ((𝜑 ∧ (𝑥 = ⟨𝑋, 𝑌⟩ ∧ 𝑦 = ⟨𝑍, 𝑊⟩)) → (𝑓 ∈ ((1^st ‘𝑦)𝐻(1^st ‘𝑥)), 𝑔 ∈ ((2^nd ‘𝑥)𝐻(2^nd ‘𝑦)) ↦ (ℎ ∈ (𝐻‘𝑥) ↦ ((𝑔(𝑥 · (2^nd ‘𝑦))ℎ)(⟨(1^st ‘𝑦), (1^st ‘𝑥)⟩ · (2^nd ‘𝑦))𝑓))) = (𝑓 ∈ (𝑍𝐻𝑋), 𝑔 ∈ (𝑌𝐻𝑊) ↦ (ℎ ∈ (𝑋𝐻𝑌) ↦ ((𝑔(⟨𝑋, 𝑌⟩ · 𝑊)ℎ)(⟨𝑍, 𝑋⟩ · 𝑊)𝑓))))
52	23, 24	opelxpd 5708	. 2 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ (𝐵 × 𝐵))
53	15, 16	opelxpd 5708	. 2 ⊢ (𝜑 → ⟨𝑍, 𝑊⟩ ∈ (𝐵 × 𝐵))
54		ovex 7438	. . . 4 ⊢ (𝑍𝐻𝑋) ∈ V
55		ovex 7438	. . . 4 ⊢ (𝑌𝐻𝑊) ∈ V
56	54, 55	mpoex 8065	. . 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 7556	1 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩(2^nd ‘𝑀)⟨𝑍, 𝑊⟩) = (𝑓 ∈ (𝑍𝐻𝑋), 𝑔 ∈ (𝑌𝐻𝑊) ↦ (ℎ ∈ (𝑋𝐻𝑌) ↦ ((𝑔(⟨𝑋, 𝑌⟩ · 𝑊)ℎ)(⟨𝑍, 𝑋⟩ · 𝑊)𝑓))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 Vcvv 3468 ⟨cop 4629 ↦ cmpt 5224 × cxp 5667 ‘cfv 6537 (class class class)co 7405 ∈ cmpo 7407 1^st c1st 7972 2^nd c2nd 7973 Basecbs 17153 Hom chom 17217 compcco 17218 Catccat 17617 Hom_f chomf 17619 Hom_Fchof 18213

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-ov 7408 df-oprab 7409 df-mpo 7410 df-1st 7974 df-2nd 7975 df-hof 18215

This theorem is referenced by: hof2val 18221

W3C validator