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Theorem hofval 17494
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 17492 . . 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 488 . . . . 5 ((𝜑𝑐 = 𝐶) → 𝑐 = 𝐶)
43fveq2d 6649 . . . 4 ((𝜑𝑐 = 𝐶) → (Homf𝑐) = (Homf𝐶))
5 fvexd 6660 . . . . 5 ((𝜑𝑐 = 𝐶) → (Base‘𝑐) ∈ V)
63fveq2d 6649 . . . . . 6 ((𝜑𝑐 = 𝐶) → (Base‘𝑐) = (Base‘𝐶))
7 hofval.b . . . . . 6 𝐵 = (Base‘𝐶)
86, 7eqtr4di 2851 . . . . 5 ((𝜑𝑐 = 𝐶) → (Base‘𝑐) = 𝐵)
9 simpr 488 . . . . . . 7 (((𝜑𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → 𝑏 = 𝐵)
109sqxpeqd 5551 . . . . . 6 (((𝜑𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → (𝑏 × 𝑏) = (𝐵 × 𝐵))
11 simplr 768 . . . . . . . . . 10 (((𝜑𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → 𝑐 = 𝐶)
1211fveq2d 6649 . . . . . . . . 9 (((𝜑𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → (Hom ‘𝑐) = (Hom ‘𝐶))
13 hofval.h . . . . . . . . 9 𝐻 = (Hom ‘𝐶)
1412, 13eqtr4di 2851 . . . . . . . 8 (((𝜑𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → (Hom ‘𝑐) = 𝐻)
1514oveqd 7152 . . . . . . 7 (((𝜑𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → ((1st𝑦)(Hom ‘𝑐)(1st𝑥)) = ((1st𝑦)𝐻(1st𝑥)))
1614oveqd 7152 . . . . . . 7 (((𝜑𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → ((2nd𝑥)(Hom ‘𝑐)(2nd𝑦)) = ((2nd𝑥)𝐻(2nd𝑦)))
1714fveq1d 6647 . . . . . . . 8 (((𝜑𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → ((Hom ‘𝑐)‘𝑥) = (𝐻𝑥))
1811fveq2d 6649 . . . . . . . . . . 11 (((𝜑𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → (comp‘𝑐) = (comp‘𝐶))
19 hofval.o . . . . . . . . . . 11 · = (comp‘𝐶)
2018, 19eqtr4di 2851 . . . . . . . . . 10 (((𝜑𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → (comp‘𝑐) = · )
2120oveqd 7152 . . . . . . . . 9 (((𝜑𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → (⟨(1st𝑦), (1st𝑥)⟩(comp‘𝑐)(2nd𝑦)) = (⟨(1st𝑦), (1st𝑥)⟩ · (2nd𝑦)))
2220oveqd 7152 . . . . . . . . . 10 (((𝜑𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → (𝑥(comp‘𝑐)(2nd𝑦)) = (𝑥 · (2nd𝑦)))
2322oveqd 7152 . . . . . . . . 9 (((𝜑𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → (𝑔(𝑥(comp‘𝑐)(2nd𝑦))) = (𝑔(𝑥 · (2nd𝑦))))
24 eqidd 2799 . . . . . . . . 9 (((𝜑𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → 𝑓 = 𝑓)
2521, 23, 24oveq123d 7156 . . . . . . . 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 5115 . . . . . . 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 7208 . . . . . 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 7208 . . . . 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 3865 . . . 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 4773 . . 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 5321 . . . 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 6753 . 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, 34syl5eq 2845 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 399   = wceq 1538  wcel 2111  Vcvv 3441  csb 3828  cop 4531  cmpt 5110   × cxp 5517  cfv 6324  (class class class)co 7135  cmpo 7137  1st c1st 7669  2nd c2nd 7670  Basecbs 16475  Hom chom 16568  compcco 16569  Catccat 16927  Homf chomf 16929  HomFchof 17490
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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-iota 6283  df-fun 6326  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-hof 17492
This theorem is referenced by:  hof1fval  17495  hof2fval  17497  hofcl  17501  hofpropd  17509
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