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Theorem hofval 18309
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 18307 . . 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 6911 . . . 4 ((𝜑𝑐 = 𝐶) → (Homf𝑐) = (Homf𝐶))
5 fvexd 6922 . . . . 5 ((𝜑𝑐 = 𝐶) → (Base‘𝑐) ∈ V)
63fveq2d 6911 . . . . . 6 ((𝜑𝑐 = 𝐶) → (Base‘𝑐) = (Base‘𝐶))
7 hofval.b . . . . . 6 𝐵 = (Base‘𝐶)
86, 7eqtr4di 2793 . . . . 5 ((𝜑𝑐 = 𝐶) → (Base‘𝑐) = 𝐵)
9 simpr 484 . . . . . . 7 (((𝜑𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → 𝑏 = 𝐵)
109sqxpeqd 5721 . . . . . 6 (((𝜑𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → (𝑏 × 𝑏) = (𝐵 × 𝐵))
11 simplr 769 . . . . . . . . . 10 (((𝜑𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → 𝑐 = 𝐶)
1211fveq2d 6911 . . . . . . . . 9 (((𝜑𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → (Hom ‘𝑐) = (Hom ‘𝐶))
13 hofval.h . . . . . . . . 9 𝐻 = (Hom ‘𝐶)
1412, 13eqtr4di 2793 . . . . . . . 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 6909 . . . . . . . 8 (((𝜑𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → ((Hom ‘𝑐)‘𝑥) = (𝐻𝑥))
1811fveq2d 6911 . . . . . . . . . . 11 (((𝜑𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → (comp‘𝑐) = (comp‘𝐶))
19 hofval.o . . . . . . . . . . 11 · = (comp‘𝐶)
2018, 19eqtr4di 2793 . . . . . . . . . 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 2736 . . . . . . . . 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 5239 . . . . . . 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 3948 . . . 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 4886 . . 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 5475 . . . 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 2787 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 1537  wcel 2106  Vcvv 3478  csb 3908  cop 4637  cmpt 5231   × cxp 5687  cfv 6563  (class class class)co 7431  cmpo 7433  1st c1st 8011  2nd c2nd 8012  Basecbs 17245  Hom chom 17309  compcco 17310  Catccat 17709  Homf chomf 17711  HomFchof 18305
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-hof 18307
This theorem is referenced by:  hof1fval  18310  hof2fval  18312  hofcl  18316  hofpropd  18324
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