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Theorem hofval 18297
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 = (HomF𝐶)
hofval.c (𝜑𝐶 ∈ Cat)
hofval.b 𝐵 = (Base‘𝐶)
hofval.h 𝐻 = (Hom ‘𝐶)
hofval.o · = (comp‘𝐶)
Assertion
Ref Expression
hofval (𝜑𝑀 = ⟨(Homf𝐶), (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ (𝐵 × 𝐵) ↦ (𝑓 ∈ ((1st𝑦)𝐻(1st𝑥)), 𝑔 ∈ ((2nd𝑥)𝐻(2nd𝑦)) ↦ ( ∈ (𝐻𝑥) ↦ ((𝑔(𝑥 · (2nd𝑦)))(⟨(1st𝑦), (1st𝑥)⟩ · (2nd𝑦))𝑓))))⟩)
Distinct variable groups:   𝑓,𝑔,,𝑥,𝑦,𝐵   𝜑,𝑓,𝑔,,𝑥,𝑦   𝐶,𝑓,𝑔,,𝑥,𝑦   𝑓,𝐻,𝑔,,𝑥,𝑦   · ,𝑓,𝑔,,𝑥,𝑦
Allowed substitution hints:   𝑀(𝑥,𝑦,𝑓,𝑔,)

Proof of Theorem hofval
Dummy variables 𝑏 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 hofval.m . 2 𝑀 = (HomF𝐶)
2 df-hof 18295 . . 3 HomF = (𝑐 ∈ Cat ↦ ⟨(Homf𝑐), (Base‘𝑐) / 𝑏(𝑥 ∈ (𝑏 × 𝑏), 𝑦 ∈ (𝑏 × 𝑏) ↦ (𝑓 ∈ ((1st𝑦)(Hom ‘𝑐)(1st𝑥)), 𝑔 ∈ ((2nd𝑥)(Hom ‘𝑐)(2nd𝑦)) ↦ ( ∈ ((Hom ‘𝑐)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝑐)(2nd𝑦)))(⟨(1st𝑦), (1st𝑥)⟩(comp‘𝑐)(2nd𝑦))𝑓))))⟩)
3 simpr 484 . . . . 5 ((𝜑𝑐 = 𝐶) → 𝑐 = 𝐶)
43fveq2d 6910 . . . 4 ((𝜑𝑐 = 𝐶) → (Homf𝑐) = (Homf𝐶))
5 fvexd 6921 . . . . 5 ((𝜑𝑐 = 𝐶) → (Base‘𝑐) ∈ V)
63fveq2d 6910 . . . . . 6 ((𝜑𝑐 = 𝐶) → (Base‘𝑐) = (Base‘𝐶))
7 hofval.b . . . . . 6 𝐵 = (Base‘𝐶)
86, 7eqtr4di 2795 . . . . 5 ((𝜑𝑐 = 𝐶) → (Base‘𝑐) = 𝐵)
9 simpr 484 . . . . . . 7 (((𝜑𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → 𝑏 = 𝐵)
109sqxpeqd 5717 . . . . . 6 (((𝜑𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → (𝑏 × 𝑏) = (𝐵 × 𝐵))
11 simplr 769 . . . . . . . . . 10 (((𝜑𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → 𝑐 = 𝐶)
1211fveq2d 6910 . . . . . . . . 9 (((𝜑𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → (Hom ‘𝑐) = (Hom ‘𝐶))
13 hofval.h . . . . . . . . 9 𝐻 = (Hom ‘𝐶)
1412, 13eqtr4di 2795 . . . . . . . 8 (((𝜑𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → (Hom ‘𝑐) = 𝐻)
1514oveqd 7448 . . . . . . 7 (((𝜑𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → ((1st𝑦)(Hom ‘𝑐)(1st𝑥)) = ((1st𝑦)𝐻(1st𝑥)))
1614oveqd 7448 . . . . . . 7 (((𝜑𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → ((2nd𝑥)(Hom ‘𝑐)(2nd𝑦)) = ((2nd𝑥)𝐻(2nd𝑦)))
1714fveq1d 6908 . . . . . . . 8 (((𝜑𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → ((Hom ‘𝑐)‘𝑥) = (𝐻𝑥))
1811fveq2d 6910 . . . . . . . . . . 11 (((𝜑𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → (comp‘𝑐) = (comp‘𝐶))
19 hofval.o . . . . . . . . . . 11 · = (comp‘𝐶)
2018, 19eqtr4di 2795 . . . . . . . . . 10 (((𝜑𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → (comp‘𝑐) = · )
2120oveqd 7448 . . . . . . . . 9 (((𝜑𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → (⟨(1st𝑦), (1st𝑥)⟩(comp‘𝑐)(2nd𝑦)) = (⟨(1st𝑦), (1st𝑥)⟩ · (2nd𝑦)))
2220oveqd 7448 . . . . . . . . . 10 (((𝜑𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → (𝑥(comp‘𝑐)(2nd𝑦)) = (𝑥 · (2nd𝑦)))
2322oveqd 7448 . . . . . . . . 9 (((𝜑𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → (𝑔(𝑥(comp‘𝑐)(2nd𝑦))) = (𝑔(𝑥 · (2nd𝑦))))
24 eqidd 2738 . . . . . . . . 9 (((𝜑𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → 𝑓 = 𝑓)
2521, 23, 24oveq123d 7452 . . . . . . . 8 (((𝜑𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → ((𝑔(𝑥(comp‘𝑐)(2nd𝑦)))(⟨(1st𝑦), (1st𝑥)⟩(comp‘𝑐)(2nd𝑦))𝑓) = ((𝑔(𝑥 · (2nd𝑦)))(⟨(1st𝑦), (1st𝑥)⟩ · (2nd𝑦))𝑓))
2617, 25mpteq12dv 5233 . . . . . . 7 (((𝜑𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → ( ∈ ((Hom ‘𝑐)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝑐)(2nd𝑦)))(⟨(1st𝑦), (1st𝑥)⟩(comp‘𝑐)(2nd𝑦))𝑓)) = ( ∈ (𝐻𝑥) ↦ ((𝑔(𝑥 · (2nd𝑦)))(⟨(1st𝑦), (1st𝑥)⟩ · (2nd𝑦))𝑓)))
2715, 16, 26mpoeq123dv 7508 . . . . . 6 (((𝜑𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → (𝑓 ∈ ((1st𝑦)(Hom ‘𝑐)(1st𝑥)), 𝑔 ∈ ((2nd𝑥)(Hom ‘𝑐)(2nd𝑦)) ↦ ( ∈ ((Hom ‘𝑐)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝑐)(2nd𝑦)))(⟨(1st𝑦), (1st𝑥)⟩(comp‘𝑐)(2nd𝑦))𝑓))) = (𝑓 ∈ ((1st𝑦)𝐻(1st𝑥)), 𝑔 ∈ ((2nd𝑥)𝐻(2nd𝑦)) ↦ ( ∈ (𝐻𝑥) ↦ ((𝑔(𝑥 · (2nd𝑦)))(⟨(1st𝑦), (1st𝑥)⟩ · (2nd𝑦))𝑓))))
2810, 10, 27mpoeq123dv 7508 . . . . 5 (((𝜑𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → (𝑥 ∈ (𝑏 × 𝑏), 𝑦 ∈ (𝑏 × 𝑏) ↦ (𝑓 ∈ ((1st𝑦)(Hom ‘𝑐)(1st𝑥)), 𝑔 ∈ ((2nd𝑥)(Hom ‘𝑐)(2nd𝑦)) ↦ ( ∈ ((Hom ‘𝑐)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝑐)(2nd𝑦)))(⟨(1st𝑦), (1st𝑥)⟩(comp‘𝑐)(2nd𝑦))𝑓)))) = (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ (𝐵 × 𝐵) ↦ (𝑓 ∈ ((1st𝑦)𝐻(1st𝑥)), 𝑔 ∈ ((2nd𝑥)𝐻(2nd𝑦)) ↦ ( ∈ (𝐻𝑥) ↦ ((𝑔(𝑥 · (2nd𝑦)))(⟨(1st𝑦), (1st𝑥)⟩ · (2nd𝑦))𝑓)))))
295, 8, 28csbied2 3936 . . . 4 ((𝜑𝑐 = 𝐶) → (Base‘𝑐) / 𝑏(𝑥 ∈ (𝑏 × 𝑏), 𝑦 ∈ (𝑏 × 𝑏) ↦ (𝑓 ∈ ((1st𝑦)(Hom ‘𝑐)(1st𝑥)), 𝑔 ∈ ((2nd𝑥)(Hom ‘𝑐)(2nd𝑦)) ↦ ( ∈ ((Hom ‘𝑐)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝑐)(2nd𝑦)))(⟨(1st𝑦), (1st𝑥)⟩(comp‘𝑐)(2nd𝑦))𝑓)))) = (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ (𝐵 × 𝐵) ↦ (𝑓 ∈ ((1st𝑦)𝐻(1st𝑥)), 𝑔 ∈ ((2nd𝑥)𝐻(2nd𝑦)) ↦ ( ∈ (𝐻𝑥) ↦ ((𝑔(𝑥 · (2nd𝑦)))(⟨(1st𝑦), (1st𝑥)⟩ · (2nd𝑦))𝑓)))))
304, 29opeq12d 4881 . . 3 ((𝜑𝑐 = 𝐶) → ⟨(Homf𝑐), (Base‘𝑐) / 𝑏(𝑥 ∈ (𝑏 × 𝑏), 𝑦 ∈ (𝑏 × 𝑏) ↦ (𝑓 ∈ ((1st𝑦)(Hom ‘𝑐)(1st𝑥)), 𝑔 ∈ ((2nd𝑥)(Hom ‘𝑐)(2nd𝑦)) ↦ ( ∈ ((Hom ‘𝑐)‘𝑥) ↦ ((𝑔(𝑥(comp‘𝑐)(2nd𝑦)))(⟨(1st𝑦), (1st𝑥)⟩(comp‘𝑐)(2nd𝑦))𝑓))))⟩ = ⟨(Homf𝐶), (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ (𝐵 × 𝐵) ↦ (𝑓 ∈ ((1st𝑦)𝐻(1st𝑥)), 𝑔 ∈ ((2nd𝑥)𝐻(2nd𝑦)) ↦ ( ∈ (𝐻𝑥) ↦ ((𝑔(𝑥 · (2nd𝑦)))(⟨(1st𝑦), (1st𝑥)⟩ · (2nd𝑦))𝑓))))⟩)
31 hofval.c . . 3 (𝜑𝐶 ∈ Cat)
32 opex 5469 . . . 4 ⟨(Homf𝐶), (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ (𝐵 × 𝐵) ↦ (𝑓 ∈ ((1st𝑦)𝐻(1st𝑥)), 𝑔 ∈ ((2nd𝑥)𝐻(2nd𝑦)) ↦ ( ∈ (𝐻𝑥) ↦ ((𝑔(𝑥 · (2nd𝑦)))(⟨(1st𝑦), (1st𝑥)⟩ · (2nd𝑦))𝑓))))⟩ ∈ V
3332a1i 11 . . 3 (𝜑 → ⟨(Homf𝐶), (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ (𝐵 × 𝐵) ↦ (𝑓 ∈ ((1st𝑦)𝐻(1st𝑥)), 𝑔 ∈ ((2nd𝑥)𝐻(2nd𝑦)) ↦ ( ∈ (𝐻𝑥) ↦ ((𝑔(𝑥 · (2nd𝑦)))(⟨(1st𝑦), (1st𝑥)⟩ · (2nd𝑦))𝑓))))⟩ ∈ V)
342, 30, 31, 33fvmptd2 7024 . 2 (𝜑 → (HomF𝐶) = ⟨(Homf𝐶), (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ (𝐵 × 𝐵) ↦ (𝑓 ∈ ((1st𝑦)𝐻(1st𝑥)), 𝑔 ∈ ((2nd𝑥)𝐻(2nd𝑦)) ↦ ( ∈ (𝐻𝑥) ↦ ((𝑔(𝑥 · (2nd𝑦)))(⟨(1st𝑦), (1st𝑥)⟩ · (2nd𝑦))𝑓))))⟩)
351, 34eqtrid 2789 1 (𝜑𝑀 = ⟨(Homf𝐶), (𝑥 ∈ (𝐵 × 𝐵), 𝑦 ∈ (𝐵 × 𝐵) ↦ (𝑓 ∈ ((1st𝑦)𝐻(1st𝑥)), 𝑔 ∈ ((2nd𝑥)𝐻(2nd𝑦)) ↦ ( ∈ (𝐻𝑥) ↦ ((𝑔(𝑥 · (2nd𝑦)))(⟨(1st𝑦), (1st𝑥)⟩ · (2nd𝑦))𝑓))))⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  Vcvv 3480  csb 3899  cop 4632  cmpt 5225   × cxp 5683  cfv 6561  (class class class)co 7431  cmpo 7433  1st c1st 8012  2nd c2nd 8013  Basecbs 17247  Hom chom 17308  compcco 17309  Catccat 17707  Homf chomf 17709  HomFchof 18293
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-hof 18295
This theorem is referenced by:  hof1fval  18298  hof2fval  18300  hofcl  18304  hofpropd  18312
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