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Theorem hofval 18303

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 18301	. . 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 489	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 = 𝐶) → 𝑐 = 𝐶)
4	3	fveq2d 6885	. . . 4 ⊢ ((𝜑 ∧ 𝑐 = 𝐶) → (Hom_f ‘𝑐) = (Hom_f ‘𝐶))
5		fvexd 6896	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 = 𝐶) → (Base‘𝑐) ∈ V)
6	3	fveq2d 6885	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 = 𝐶) → (Base‘𝑐) = (Base‘𝐶))
7		hofval.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐶)
8	6, 7	eqtr4di 2816	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 = 𝐶) → (Base‘𝑐) = 𝐵)
9		simpr 489	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → 𝑏 = 𝐵)
10	9	sqxpeqd 5693	. . . . . 6 ⊢ (((𝜑 ∧ 𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → (𝑏 × 𝑏) = (𝐵 × 𝐵))
11		simplr 780	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → 𝑐 = 𝐶)
12	11	fveq2d 6885	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → (Hom ‘𝑐) = (Hom ‘𝐶))
13		hofval.h	. . . . . . . . 9 ⊢ 𝐻 = (Hom ‘𝐶)
14	12, 13	eqtr4di 2816	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → (Hom ‘𝑐) = 𝐻)
15	14	oveqd 7427	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → ((1^st ‘𝑦)(Hom ‘𝑐)(1^st ‘𝑥)) = ((1^st ‘𝑦)𝐻(1^st ‘𝑥)))
16	14	oveqd 7427	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → ((2^nd ‘𝑥)(Hom ‘𝑐)(2^nd ‘𝑦)) = ((2^nd ‘𝑥)𝐻(2^nd ‘𝑦)))
17	14	fveq1d 6883	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → ((Hom ‘𝑐)‘𝑥) = (𝐻‘𝑥))
18	11	fveq2d 6885	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → (comp‘𝑐) = (comp‘𝐶))
19		hofval.o	. . . . . . . . . . 11 ⊢ · = (comp‘𝐶)
20	18, 19	eqtr4di 2816	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → (comp‘𝑐) = · )
21	20	oveqd 7427	. . . . . . . . 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 7427	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → (𝑥(comp‘𝑐)(2^nd ‘𝑦)) = (𝑥 · (2^nd ‘𝑦)))
23	22	oveqd 7427	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → (𝑔(𝑥(comp‘𝑐)(2^nd ‘𝑦))ℎ) = (𝑔(𝑥 · (2^nd ‘𝑦))ℎ))
24		eqidd 2764	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → 𝑓 = 𝑓)
25	21, 23, 24	oveq123d 7431	. . . . . . . 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 5198	. . . . . . 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 7485	. . . . . 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 7485	. . . . 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 3890	. . . 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 4846	. . 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 5445	. . . 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 6998	. 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 2810	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 400 = wceq 1570 ∈ wcel 2143 Vcvv 3455 ⦋csb 3853 ⟨cop 4595 ↦ cmpt 5192 × cxp 5659 ‘cfv 6536 (class class class)co 7410 ∈ cmpo 7412 1^st c1st 7980 2^nd c2nd 7981 Basecbs 17264 Hom chom 17316 compcco 17317 Catccat 17715 Hom_f chomf 17717 Hom_Fchof 18299

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-sep 5257 ax-nul 5269 ax-pr 5404

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-ral 3080 df-rex 3090 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-nul 4287 df-if 4488 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4873 df-br 5110 df-opab 5174 df-mpt 5193 df-id 5556 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-iota 6492 df-fun 6538 df-fv 6544 df-ov 7413 df-oprab 7414 df-mpo 7415 df-hof 18301

This theorem is referenced by: hof1fval 18304 hof2fval 18306 hofcl 18310 hofpropd 18318

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