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Theorem hofval 18067
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 18065 . . 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 486 . . . . 5 ((𝜑𝑐 = 𝐶) → 𝑐 = 𝐶)
43fveq2d 6833 . . . 4 ((𝜑𝑐 = 𝐶) → (Homf𝑐) = (Homf𝐶))
5 fvexd 6844 . . . . 5 ((𝜑𝑐 = 𝐶) → (Base‘𝑐) ∈ V)
63fveq2d 6833 . . . . . 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 486 . . . . . . 7 (((𝜑𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → 𝑏 = 𝐵)
109sqxpeqd 5656 . . . . . 6 (((𝜑𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → (𝑏 × 𝑏) = (𝐵 × 𝐵))
11 simplr 767 . . . . . . . . . 10 (((𝜑𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → 𝑐 = 𝐶)
1211fveq2d 6833 . . . . . . . . 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 7358 . . . . . . 7 (((𝜑𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → ((1st𝑦)(Hom ‘𝑐)(1st𝑥)) = ((1st𝑦)𝐻(1st𝑥)))
1614oveqd 7358 . . . . . . 7 (((𝜑𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → ((2nd𝑥)(Hom ‘𝑐)(2nd𝑦)) = ((2nd𝑥)𝐻(2nd𝑦)))
1714fveq1d 6831 . . . . . . . 8 (((𝜑𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → ((Hom ‘𝑐)‘𝑥) = (𝐻𝑥))
1811fveq2d 6833 . . . . . . . . . . 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 7358 . . . . . . . . 9 (((𝜑𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → (⟨(1st𝑦), (1st𝑥)⟩(comp‘𝑐)(2nd𝑦)) = (⟨(1st𝑦), (1st𝑥)⟩ · (2nd𝑦)))
2220oveqd 7358 . . . . . . . . . 10 (((𝜑𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → (𝑥(comp‘𝑐)(2nd𝑦)) = (𝑥 · (2nd𝑦)))
2322oveqd 7358 . . . . . . . . 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 7362 . . . . . . . 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 5187 . . . . . . 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 7416 . . . . . 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 7416 . . . . 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 3886 . . . 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 4829 . . 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 5413 . . . 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 6943 . 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 397   = wceq 1541  wcel 2106  Vcvv 3442  csb 3846  cop 4583  cmpt 5179   × cxp 5622  cfv 6483  (class class class)co 7341  cmpo 7343  1st c1st 7901  2nd c2nd 7902  Basecbs 17009  Hom chom 17070  compcco 17071  Catccat 17470  Homf chomf 17472  HomFchof 18063
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2708  ax-sep 5247  ax-nul 5254  ax-pr 5376
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3405  df-v 3444  df-sbc 3731  df-csb 3847  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4274  df-if 4478  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4857  df-br 5097  df-opab 5159  df-mpt 5180  df-id 5522  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-iota 6435  df-fun 6485  df-fv 6491  df-ov 7344  df-oprab 7345  df-mpo 7346  df-hof 18065
This theorem is referenced by:  hof1fval  18068  hof2fval  18070  hofcl  18074  hofpropd  18082
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