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Theorem hofval 17970

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 17968	. . 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 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 = 𝐶) → 𝑐 = 𝐶)
4	3	fveq2d 6778	. . . 4 ⊢ ((𝜑 ∧ 𝑐 = 𝐶) → (Hom_f ‘𝑐) = (Hom_f ‘𝐶))
5		fvexd 6789	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 = 𝐶) → (Base‘𝑐) ∈ V)
6	3	fveq2d 6778	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 = 𝐶) → (Base‘𝑐) = (Base‘𝐶))
7		hofval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐶)
8	6, 7	eqtr4di 2796	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 = 𝐶) → (Base‘𝑐) = 𝐵)
9		simpr 485	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → 𝑏 = 𝐵)
10	9	sqxpeqd 5621	. . . . . 6 ⊢ (((𝜑 ∧ 𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → (𝑏 × 𝑏) = (𝐵 × 𝐵))
11		simplr 766	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → 𝑐 = 𝐶)
12	11	fveq2d 6778	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → (Hom ‘𝑐) = (Hom ‘𝐶))
13		hofval.h	. . . . . . . . 9 ⊢ 𝐻 = (Hom ‘𝐶)
14	12, 13	eqtr4di 2796	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → (Hom ‘𝑐) = 𝐻)
15	14	oveqd 7292	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → ((1^st ‘𝑦)(Hom ‘𝑐)(1^st ‘𝑥)) = ((1^st ‘𝑦)𝐻(1^st ‘𝑥)))
16	14	oveqd 7292	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → ((2^nd ‘𝑥)(Hom ‘𝑐)(2^nd ‘𝑦)) = ((2^nd ‘𝑥)𝐻(2^nd ‘𝑦)))
17	14	fveq1d 6776	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → ((Hom ‘𝑐)‘𝑥) = (𝐻‘𝑥))
18	11	fveq2d 6778	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → (comp‘𝑐) = (comp‘𝐶))
19		hofval.o	. . . . . . . . . . 11 ⊢ · = (comp‘𝐶)
20	18, 19	eqtr4di 2796	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → (comp‘𝑐) = · )
21	20	oveqd 7292	. . . . . . . . 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 7292	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → (𝑥(comp‘𝑐)(2^nd ‘𝑦)) = (𝑥 · (2^nd ‘𝑦)))
23	22	oveqd 7292	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → (𝑔(𝑥(comp‘𝑐)(2^nd ‘𝑦))ℎ) = (𝑔(𝑥 · (2^nd ‘𝑦))ℎ))
24		eqidd 2739	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → 𝑓 = 𝑓)
25	21, 23, 24	oveq123d 7296	. . . . . . . 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 5165	. . . . . . 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 7350	. . . . . 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 7350	. . . . 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 3872	. . . 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 4812	. . 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 5379	. . . 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 6883	. 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 2790	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 396 = wceq 1539 ∈ wcel 2106 Vcvv 3432 ⦋csb 3832 ⟨cop 4567 ↦ cmpt 5157 × cxp 5587 ‘cfv 6433 (class class class)co 7275 ∈ cmpo 7277 1^st c1st 7829 2^nd c2nd 7830 Basecbs 16912 Hom chom 16973 compcco 16974 Catccat 17373 Hom_f chomf 17375 Hom_Fchof 17966

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 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 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-iota 6391 df-fun 6435 df-fv 6441 df-ov 7278 df-oprab 7279 df-mpo 7280 df-hof 17968

This theorem is referenced by: hof1fval 17971 hof2fval 17973 hofcl 17977 hofpropd 17985

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