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Theorem hofval 18277

Description: Value of the Hom functor, which is a bifunctor (a functor of two arguments), contravariant in the first argument and covariant in the second, from (oppCat‘𝐶) × 𝐶 to SetCat, whose object part is the hom-function Hom, and with morphism part given by pre- and post-composition. (Contributed by Mario Carneiro, 15-Jan-2017.)

**Hypotheses**
Ref	Expression
hofval.m	⊢ 𝑀 = (Hom_F‘𝐶)
hofval.c	⊢ (𝜑 → 𝐶 ∈ Cat)
hofval.b	⊢ 𝐵 = (Base‘𝐶)
hofval.h	⊢ 𝐻 = (Hom ‘𝐶)
hofval.o	⊢ · = (comp‘𝐶)

**Assertion**
Ref	Expression
hofval	⊢ (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))))⟩)

Distinct variable groups: 𝑓,𝑔,ℎ,𝑥,𝑦,𝐵 𝜑,𝑓,𝑔,ℎ,𝑥,𝑦 𝐶,𝑓,𝑔,ℎ,𝑥,𝑦 𝑓,𝐻,𝑔,ℎ,𝑥,𝑦 · ,𝑓,𝑔,ℎ,𝑥,𝑦 Allowed substitution hints: 𝑀(𝑥,𝑦,𝑓,𝑔,ℎ)

Proof of Theorem hofval Dummy variables 𝑏 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		hofval.m	. 2 ⊢ 𝑀 = (Hom_F‘𝐶)
2		df-hof 18275	. . 3 ⊢ Hom_F = (𝑐 ∈ Cat ↦ ⟨(Hom_f ‘𝑐), ⦋(Base‘𝑐) / 𝑏⦌(𝑥 ∈ (𝑏 × 𝑏), 𝑦 ∈ (𝑏 × 𝑏) ↦ (𝑓 ∈ ((1^st ‘𝑦)(Hom ‘𝑐)(1^st ‘𝑥)), 𝑔 ∈ ((2^nd ‘𝑥)(Hom ‘𝑐)(2^nd ‘𝑦)) ↦ (ℎ ∈ ((Hom ‘𝑐)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝑐)(2^nd ‘𝑦))ℎ)(⟨(1^st ‘𝑦), (1^st ‘𝑥)⟩(comp‘𝑐)(2^nd ‘𝑦))𝑓))))⟩)
3		simpr 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 = 𝐶) → 𝑐 = 𝐶)
4	3	fveq2d 6905	. . . 4 ⊢ ((𝜑 ∧ 𝑐 = 𝐶) → (Hom_f ‘𝑐) = (Hom_f ‘𝐶))
5		fvexd 6916	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 = 𝐶) → (Base‘𝑐) ∈ V)
6	3	fveq2d 6905	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 = 𝐶) → (Base‘𝑐) = (Base‘𝐶))
7		hofval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐶)
8	6, 7	eqtr4di 2784	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 = 𝐶) → (Base‘𝑐) = 𝐵)
9		simpr 483	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → 𝑏 = 𝐵)
10	9	sqxpeqd 5714	. . . . . 6 ⊢ (((𝜑 ∧ 𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → (𝑏 × 𝑏) = (𝐵 × 𝐵))
11		simplr 767	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → 𝑐 = 𝐶)
12	11	fveq2d 6905	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → (Hom ‘𝑐) = (Hom ‘𝐶))
13		hofval.h	. . . . . . . . 9 ⊢ 𝐻 = (Hom ‘𝐶)
14	12, 13	eqtr4di 2784	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → (Hom ‘𝑐) = 𝐻)
15	14	oveqd 7441	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → ((1^st ‘𝑦)(Hom ‘𝑐)(1^st ‘𝑥)) = ((1^st ‘𝑦)𝐻(1^st ‘𝑥)))
16	14	oveqd 7441	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → ((2^nd ‘𝑥)(Hom ‘𝑐)(2^nd ‘𝑦)) = ((2^nd ‘𝑥)𝐻(2^nd ‘𝑦)))
17	14	fveq1d 6903	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → ((Hom ‘𝑐)‘𝑥) = (𝐻‘𝑥))
18	11	fveq2d 6905	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → (comp‘𝑐) = (comp‘𝐶))
19		hofval.o	. . . . . . . . . . 11 ⊢ · = (comp‘𝐶)
20	18, 19	eqtr4di 2784	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → (comp‘𝑐) = · )
21	20	oveqd 7441	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → (⟨(1^st ‘𝑦), (1^st ‘𝑥)⟩(comp‘𝑐)(2^nd ‘𝑦)) = (⟨(1^st ‘𝑦), (1^st ‘𝑥)⟩ · (2^nd ‘𝑦)))
22	20	oveqd 7441	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → (𝑥(comp‘𝑐)(2^nd ‘𝑦)) = (𝑥 · (2^nd ‘𝑦)))
23	22	oveqd 7441	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → (𝑔(𝑥(comp‘𝑐)(2^nd ‘𝑦))ℎ) = (𝑔(𝑥 · (2^nd ‘𝑦))ℎ))
24		eqidd 2727	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → 𝑓 = 𝑓)
25	21, 23, 24	oveq123d 7445	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → ((𝑔(𝑥(comp‘𝑐)(2^nd ‘𝑦))ℎ)(⟨(1^st ‘𝑦), (1^st ‘𝑥)⟩(comp‘𝑐)(2^nd ‘𝑦))𝑓) = ((𝑔(𝑥 · (2^nd ‘𝑦))ℎ)(⟨(1^st ‘𝑦), (1^st ‘𝑥)⟩ · (2^nd ‘𝑦))𝑓))
26	17, 25	mpteq12dv 5244	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → (ℎ ∈ ((Hom ‘𝑐)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝑐)(2^nd ‘𝑦))ℎ)(⟨(1^st ‘𝑦), (1^st ‘𝑥)⟩(comp‘𝑐)(2^nd ‘𝑦))𝑓)) = (ℎ ∈ (𝐻‘𝑥) ↦ ((𝑔(𝑥 · (2^nd ‘𝑦))ℎ)(⟨(1^st ‘𝑦), (1^st ‘𝑥)⟩ · (2^nd ‘𝑦))𝑓)))
27	15, 16, 26	mpoeq123dv 7500	. . . . . 6 ⊢ (((𝜑 ∧ 𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → (𝑓 ∈ ((1^st ‘𝑦)(Hom ‘𝑐)(1^st ‘𝑥)), 𝑔 ∈ ((2^nd ‘𝑥)(Hom ‘𝑐)(2^nd ‘𝑦)) ↦ (ℎ ∈ ((Hom ‘𝑐)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝑐)(2^nd ‘𝑦))ℎ)(⟨(1^st ‘𝑦), (1^st ‘𝑥)⟩(comp‘𝑐)(2^nd ‘𝑦))𝑓))) = (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))))
28	10, 10, 27	mpoeq123dv 7500	. . . . 5 ⊢ (((𝜑 ∧ 𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → (𝑥 ∈ (𝑏 × 𝑏), 𝑦 ∈ (𝑏 × 𝑏) ↦ (𝑓 ∈ ((1^st ‘𝑦)(Hom ‘𝑐)(1^st ‘𝑥)), 𝑔 ∈ ((2^nd ‘𝑥)(Hom ‘𝑐)(2^nd ‘𝑦)) ↦ (ℎ ∈ ((Hom ‘𝑐)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝑐)(2^nd ‘𝑦))ℎ)(⟨(1^st ‘𝑦), (1^st ‘𝑥)⟩(comp‘𝑐)(2^nd ‘𝑦))𝑓)))) = (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)))))
29	5, 8, 28	csbied2 3932	. . . 4 ⊢ ((𝜑 ∧ 𝑐 = 𝐶) → ⦋(Base‘𝑐) / 𝑏⦌(𝑥 ∈ (𝑏 × 𝑏), 𝑦 ∈ (𝑏 × 𝑏) ↦ (𝑓 ∈ ((1^st ‘𝑦)(Hom ‘𝑐)(1^st ‘𝑥)), 𝑔 ∈ ((2^nd ‘𝑥)(Hom ‘𝑐)(2^nd ‘𝑦)) ↦ (ℎ ∈ ((Hom ‘𝑐)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝑐)(2^nd ‘𝑦))ℎ)(⟨(1^st ‘𝑦), (1^st ‘𝑥)⟩(comp‘𝑐)(2^nd ‘𝑦))𝑓)))) = (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)))))
30	4, 29	opeq12d 4887	. . 3 ⊢ ((𝜑 ∧ 𝑐 = 𝐶) → ⟨(Hom_f ‘𝑐), ⦋(Base‘𝑐) / 𝑏⦌(𝑥 ∈ (𝑏 × 𝑏), 𝑦 ∈ (𝑏 × 𝑏) ↦ (𝑓 ∈ ((1^st ‘𝑦)(Hom ‘𝑐)(1^st ‘𝑥)), 𝑔 ∈ ((2^nd ‘𝑥)(Hom ‘𝑐)(2^nd ‘𝑦)) ↦ (ℎ ∈ ((Hom ‘𝑐)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝑐)(2^nd ‘𝑦))ℎ)(⟨(1^st ‘𝑦), (1^st ‘𝑥)⟩(comp‘𝑐)(2^nd ‘𝑦))𝑓))))⟩ = ⟨(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))))⟩)
31		hofval.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ Cat)
32		opex 5470	. . . 4 ⊢ ⟨(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))))⟩ ∈ V
33	32	a1i 11	. . 3 ⊢ (𝜑 → ⟨(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))))⟩ ∈ V)
34	2, 30, 31, 33	fvmptd2 7017	. 2 ⊢ (𝜑 → (Hom_F‘𝐶) = ⟨(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))))⟩)
35	1, 34	eqtrid 2778	1 ⊢ (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))))⟩)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 = wceq 1534 ∈ wcel 2099 Vcvv 3462 ⦋csb 3892 ⟨cop 4639 ↦ cmpt 5236 × cxp 5680 ‘cfv 6554 (class class class)co 7424 ∈ cmpo 7426 1^st c1st 8001 2^nd c2nd 8002 Basecbs 17213 Hom chom 17277 compcco 17278 Catccat 17677 Hom_f chomf 17679 Hom_Fchof 18273

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pr 5433

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-ss 3964 df-nul 4326 df-if 4534 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-br 5154 df-opab 5216 df-mpt 5237 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-iota 6506 df-fun 6556 df-fv 6562 df-ov 7427 df-oprab 7428 df-mpo 7429 df-hof 18275

This theorem is referenced by: hof1fval 18278 hof2fval 18280 hofcl 18284 hofpropd 18292

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