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Theorem hofval 17372
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 17370 . . 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 477 . . . . 5 ((𝜑𝑐 = 𝐶) → 𝑐 = 𝐶)
43fveq2d 6500 . . . 4 ((𝜑𝑐 = 𝐶) → (Homf𝑐) = (Homf𝐶))
5 fvexd 6511 . . . . 5 ((𝜑𝑐 = 𝐶) → (Base‘𝑐) ∈ V)
63fveq2d 6500 . . . . . 6 ((𝜑𝑐 = 𝐶) → (Base‘𝑐) = (Base‘𝐶))
7 hofval.b . . . . . 6 𝐵 = (Base‘𝐶)
86, 7syl6eqr 2825 . . . . 5 ((𝜑𝑐 = 𝐶) → (Base‘𝑐) = 𝐵)
9 simpr 477 . . . . . . 7 (((𝜑𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → 𝑏 = 𝐵)
109sqxpeqd 5435 . . . . . 6 (((𝜑𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → (𝑏 × 𝑏) = (𝐵 × 𝐵))
11 simplr 757 . . . . . . . . . 10 (((𝜑𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → 𝑐 = 𝐶)
1211fveq2d 6500 . . . . . . . . 9 (((𝜑𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → (Hom ‘𝑐) = (Hom ‘𝐶))
13 hofval.h . . . . . . . . 9 𝐻 = (Hom ‘𝐶)
1412, 13syl6eqr 2825 . . . . . . . 8 (((𝜑𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → (Hom ‘𝑐) = 𝐻)
1514oveqd 6991 . . . . . . 7 (((𝜑𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → ((1st𝑦)(Hom ‘𝑐)(1st𝑥)) = ((1st𝑦)𝐻(1st𝑥)))
1614oveqd 6991 . . . . . . 7 (((𝜑𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → ((2nd𝑥)(Hom ‘𝑐)(2nd𝑦)) = ((2nd𝑥)𝐻(2nd𝑦)))
1714fveq1d 6498 . . . . . . . 8 (((𝜑𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → ((Hom ‘𝑐)‘𝑥) = (𝐻𝑥))
1811fveq2d 6500 . . . . . . . . . . 11 (((𝜑𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → (comp‘𝑐) = (comp‘𝐶))
19 hofval.o . . . . . . . . . . 11 · = (comp‘𝐶)
2018, 19syl6eqr 2825 . . . . . . . . . 10 (((𝜑𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → (comp‘𝑐) = · )
2120oveqd 6991 . . . . . . . . 9 (((𝜑𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → (⟨(1st𝑦), (1st𝑥)⟩(comp‘𝑐)(2nd𝑦)) = (⟨(1st𝑦), (1st𝑥)⟩ · (2nd𝑦)))
2220oveqd 6991 . . . . . . . . . 10 (((𝜑𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → (𝑥(comp‘𝑐)(2nd𝑦)) = (𝑥 · (2nd𝑦)))
2322oveqd 6991 . . . . . . . . 9 (((𝜑𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → (𝑔(𝑥(comp‘𝑐)(2nd𝑦))) = (𝑔(𝑥 · (2nd𝑦))))
24 eqidd 2772 . . . . . . . . 9 (((𝜑𝑐 = 𝐶) ∧ 𝑏 = 𝐵) → 𝑓 = 𝑓)
2521, 23, 24oveq123d 6995 . . . . . . . 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 5008 . . . . . . 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 7045 . . . . . 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 7045 . . . . 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 3809 . . . 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 4681 . . 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 5209 . . . 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 6600 . 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 2819 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 387   = wceq 1508  wcel 2051  Vcvv 3408  csb 3779  cop 4441  cmpt 5004   × cxp 5401  cfv 6185  (class class class)co 6974  cmpo 6976  1st c1st 7497  2nd c2nd 7498  Basecbs 16337  Hom chom 16430  compcco 16431  Catccat 16805  Homf chomf 16807  HomFchof 17368
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2743  ax-sep 5056  ax-nul 5063  ax-pr 5182
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ral 3086  df-rex 3087  df-rab 3090  df-v 3410  df-sbc 3675  df-csb 3780  df-dif 3825  df-un 3827  df-in 3829  df-ss 3836  df-nul 4173  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4709  df-br 4926  df-opab 4988  df-mpt 5005  df-id 5308  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-iota 6149  df-fun 6187  df-fv 6193  df-ov 6977  df-oprab 6978  df-mpo 6979  df-hof 17370
This theorem is referenced by:  hof1fval  17373  hof2fval  17375  hofcl  17379  hofpropd  17387
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