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Theorem fucval 16441
 Description: Value of the functor category. (Contributed by Mario Carneiro, 6-Jan-2017.)
Hypotheses
Ref Expression
fucval.q 𝑄 = (𝐶 FuncCat 𝐷)
fucval.b 𝐵 = (𝐶 Func 𝐷)
fucval.n 𝑁 = (𝐶 Nat 𝐷)
fucval.a 𝐴 = (Base‘𝐶)
fucval.o · = (comp‘𝐷)
fucval.c (𝜑𝐶 ∈ Cat)
fucval.d (𝜑𝐷 ∈ Cat)
fucval.x (𝜑 = (𝑣 ∈ (𝐵 × 𝐵), 𝐵(1st𝑣) / 𝑓(2nd𝑣) / 𝑔(𝑏 ∈ (𝑔𝑁), 𝑎 ∈ (𝑓𝑁𝑔) ↦ (𝑥𝐴 ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩ · ((1st)‘𝑥))(𝑎𝑥))))))
Assertion
Ref Expression
fucval (𝜑𝑄 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝑁⟩, ⟨(comp‘ndx), ⟩})
Distinct variable groups:   𝑣,,𝐵   𝑎,𝑏,𝑓,𝑔,,𝑣,𝑥,𝜑   𝐶,𝑎,𝑏,𝑓,𝑔,,𝑣,𝑥   𝐷,𝑎,𝑏,𝑓,𝑔,,𝑣,𝑥
Allowed substitution hints:   𝐴(𝑥,𝑣,𝑓,𝑔,,𝑎,𝑏)   𝐵(𝑥,𝑓,𝑔,𝑎,𝑏)   𝑄(𝑥,𝑣,𝑓,𝑔,,𝑎,𝑏)   (𝑥,𝑣,𝑓,𝑔,,𝑎,𝑏)   · (𝑥,𝑣,𝑓,𝑔,,𝑎,𝑏)   𝑁(𝑥,𝑣,𝑓,𝑔,,𝑎,𝑏)

Proof of Theorem fucval
Dummy variables 𝑡 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fucval.q . 2 𝑄 = (𝐶 FuncCat 𝐷)
2 df-fuc 16427 . . . 4 FuncCat = (𝑡 ∈ Cat, 𝑢 ∈ Cat ↦ {⟨(Base‘ndx), (𝑡 Func 𝑢)⟩, ⟨(Hom ‘ndx), (𝑡 Nat 𝑢)⟩, ⟨(comp‘ndx), (𝑣 ∈ ((𝑡 Func 𝑢) × (𝑡 Func 𝑢)), ∈ (𝑡 Func 𝑢) ↦ (1st𝑣) / 𝑓(2nd𝑣) / 𝑔(𝑏 ∈ (𝑔(𝑡 Nat 𝑢)), 𝑎 ∈ (𝑓(𝑡 Nat 𝑢)𝑔) ↦ (𝑥 ∈ (Base‘𝑡) ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝑢)((1st)‘𝑥))(𝑎𝑥)))))⟩})
32a1i 11 . . 3 (𝜑 → FuncCat = (𝑡 ∈ Cat, 𝑢 ∈ Cat ↦ {⟨(Base‘ndx), (𝑡 Func 𝑢)⟩, ⟨(Hom ‘ndx), (𝑡 Nat 𝑢)⟩, ⟨(comp‘ndx), (𝑣 ∈ ((𝑡 Func 𝑢) × (𝑡 Func 𝑢)), ∈ (𝑡 Func 𝑢) ↦ (1st𝑣) / 𝑓(2nd𝑣) / 𝑔(𝑏 ∈ (𝑔(𝑡 Nat 𝑢)), 𝑎 ∈ (𝑓(𝑡 Nat 𝑢)𝑔) ↦ (𝑥 ∈ (Base‘𝑡) ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝑢)((1st)‘𝑥))(𝑎𝑥)))))⟩}))
4 simprl 790 . . . . . . 7 ((𝜑 ∧ (𝑡 = 𝐶𝑢 = 𝐷)) → 𝑡 = 𝐶)
5 simprr 792 . . . . . . 7 ((𝜑 ∧ (𝑡 = 𝐶𝑢 = 𝐷)) → 𝑢 = 𝐷)
64, 5oveq12d 6567 . . . . . 6 ((𝜑 ∧ (𝑡 = 𝐶𝑢 = 𝐷)) → (𝑡 Func 𝑢) = (𝐶 Func 𝐷))
7 fucval.b . . . . . 6 𝐵 = (𝐶 Func 𝐷)
86, 7syl6eqr 2662 . . . . 5 ((𝜑 ∧ (𝑡 = 𝐶𝑢 = 𝐷)) → (𝑡 Func 𝑢) = 𝐵)
98opeq2d 4347 . . . 4 ((𝜑 ∧ (𝑡 = 𝐶𝑢 = 𝐷)) → ⟨(Base‘ndx), (𝑡 Func 𝑢)⟩ = ⟨(Base‘ndx), 𝐵⟩)
104, 5oveq12d 6567 . . . . . 6 ((𝜑 ∧ (𝑡 = 𝐶𝑢 = 𝐷)) → (𝑡 Nat 𝑢) = (𝐶 Nat 𝐷))
11 fucval.n . . . . . 6 𝑁 = (𝐶 Nat 𝐷)
1210, 11syl6eqr 2662 . . . . 5 ((𝜑 ∧ (𝑡 = 𝐶𝑢 = 𝐷)) → (𝑡 Nat 𝑢) = 𝑁)
1312opeq2d 4347 . . . 4 ((𝜑 ∧ (𝑡 = 𝐶𝑢 = 𝐷)) → ⟨(Hom ‘ndx), (𝑡 Nat 𝑢)⟩ = ⟨(Hom ‘ndx), 𝑁⟩)
148sqxpeqd 5065 . . . . . . 7 ((𝜑 ∧ (𝑡 = 𝐶𝑢 = 𝐷)) → ((𝑡 Func 𝑢) × (𝑡 Func 𝑢)) = (𝐵 × 𝐵))
1512oveqd 6566 . . . . . . . . . 10 ((𝜑 ∧ (𝑡 = 𝐶𝑢 = 𝐷)) → (𝑔(𝑡 Nat 𝑢)) = (𝑔𝑁))
1612oveqd 6566 . . . . . . . . . 10 ((𝜑 ∧ (𝑡 = 𝐶𝑢 = 𝐷)) → (𝑓(𝑡 Nat 𝑢)𝑔) = (𝑓𝑁𝑔))
174fveq2d 6107 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑡 = 𝐶𝑢 = 𝐷)) → (Base‘𝑡) = (Base‘𝐶))
18 fucval.a . . . . . . . . . . . 12 𝐴 = (Base‘𝐶)
1917, 18syl6eqr 2662 . . . . . . . . . . 11 ((𝜑 ∧ (𝑡 = 𝐶𝑢 = 𝐷)) → (Base‘𝑡) = 𝐴)
205fveq2d 6107 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑡 = 𝐶𝑢 = 𝐷)) → (comp‘𝑢) = (comp‘𝐷))
21 fucval.o . . . . . . . . . . . . . 14 · = (comp‘𝐷)
2220, 21syl6eqr 2662 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑡 = 𝐶𝑢 = 𝐷)) → (comp‘𝑢) = · )
2322oveqd 6566 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑡 = 𝐶𝑢 = 𝐷)) → (⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝑢)((1st)‘𝑥)) = (⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩ · ((1st)‘𝑥)))
2423oveqd 6566 . . . . . . . . . . 11 ((𝜑 ∧ (𝑡 = 𝐶𝑢 = 𝐷)) → ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝑢)((1st)‘𝑥))(𝑎𝑥)) = ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩ · ((1st)‘𝑥))(𝑎𝑥)))
2519, 24mpteq12dv 4663 . . . . . . . . . 10 ((𝜑 ∧ (𝑡 = 𝐶𝑢 = 𝐷)) → (𝑥 ∈ (Base‘𝑡) ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝑢)((1st)‘𝑥))(𝑎𝑥))) = (𝑥𝐴 ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩ · ((1st)‘𝑥))(𝑎𝑥))))
2615, 16, 25mpt2eq123dv 6615 . . . . . . . . 9 ((𝜑 ∧ (𝑡 = 𝐶𝑢 = 𝐷)) → (𝑏 ∈ (𝑔(𝑡 Nat 𝑢)), 𝑎 ∈ (𝑓(𝑡 Nat 𝑢)𝑔) ↦ (𝑥 ∈ (Base‘𝑡) ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝑢)((1st)‘𝑥))(𝑎𝑥)))) = (𝑏 ∈ (𝑔𝑁), 𝑎 ∈ (𝑓𝑁𝑔) ↦ (𝑥𝐴 ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩ · ((1st)‘𝑥))(𝑎𝑥)))))
2726csbeq2dv 3944 . . . . . . . 8 ((𝜑 ∧ (𝑡 = 𝐶𝑢 = 𝐷)) → (2nd𝑣) / 𝑔(𝑏 ∈ (𝑔(𝑡 Nat 𝑢)), 𝑎 ∈ (𝑓(𝑡 Nat 𝑢)𝑔) ↦ (𝑥 ∈ (Base‘𝑡) ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝑢)((1st)‘𝑥))(𝑎𝑥)))) = (2nd𝑣) / 𝑔(𝑏 ∈ (𝑔𝑁), 𝑎 ∈ (𝑓𝑁𝑔) ↦ (𝑥𝐴 ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩ · ((1st)‘𝑥))(𝑎𝑥)))))
2827csbeq2dv 3944 . . . . . . 7 ((𝜑 ∧ (𝑡 = 𝐶𝑢 = 𝐷)) → (1st𝑣) / 𝑓(2nd𝑣) / 𝑔(𝑏 ∈ (𝑔(𝑡 Nat 𝑢)), 𝑎 ∈ (𝑓(𝑡 Nat 𝑢)𝑔) ↦ (𝑥 ∈ (Base‘𝑡) ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝑢)((1st)‘𝑥))(𝑎𝑥)))) = (1st𝑣) / 𝑓(2nd𝑣) / 𝑔(𝑏 ∈ (𝑔𝑁), 𝑎 ∈ (𝑓𝑁𝑔) ↦ (𝑥𝐴 ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩ · ((1st)‘𝑥))(𝑎𝑥)))))
2914, 8, 28mpt2eq123dv 6615 . . . . . 6 ((𝜑 ∧ (𝑡 = 𝐶𝑢 = 𝐷)) → (𝑣 ∈ ((𝑡 Func 𝑢) × (𝑡 Func 𝑢)), ∈ (𝑡 Func 𝑢) ↦ (1st𝑣) / 𝑓(2nd𝑣) / 𝑔(𝑏 ∈ (𝑔(𝑡 Nat 𝑢)), 𝑎 ∈ (𝑓(𝑡 Nat 𝑢)𝑔) ↦ (𝑥 ∈ (Base‘𝑡) ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝑢)((1st)‘𝑥))(𝑎𝑥))))) = (𝑣 ∈ (𝐵 × 𝐵), 𝐵(1st𝑣) / 𝑓(2nd𝑣) / 𝑔(𝑏 ∈ (𝑔𝑁), 𝑎 ∈ (𝑓𝑁𝑔) ↦ (𝑥𝐴 ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩ · ((1st)‘𝑥))(𝑎𝑥))))))
30 fucval.x . . . . . . 7 (𝜑 = (𝑣 ∈ (𝐵 × 𝐵), 𝐵(1st𝑣) / 𝑓(2nd𝑣) / 𝑔(𝑏 ∈ (𝑔𝑁), 𝑎 ∈ (𝑓𝑁𝑔) ↦ (𝑥𝐴 ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩ · ((1st)‘𝑥))(𝑎𝑥))))))
3130adantr 480 . . . . . 6 ((𝜑 ∧ (𝑡 = 𝐶𝑢 = 𝐷)) → = (𝑣 ∈ (𝐵 × 𝐵), 𝐵(1st𝑣) / 𝑓(2nd𝑣) / 𝑔(𝑏 ∈ (𝑔𝑁), 𝑎 ∈ (𝑓𝑁𝑔) ↦ (𝑥𝐴 ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩ · ((1st)‘𝑥))(𝑎𝑥))))))
3229, 31eqtr4d 2647 . . . . 5 ((𝜑 ∧ (𝑡 = 𝐶𝑢 = 𝐷)) → (𝑣 ∈ ((𝑡 Func 𝑢) × (𝑡 Func 𝑢)), ∈ (𝑡 Func 𝑢) ↦ (1st𝑣) / 𝑓(2nd𝑣) / 𝑔(𝑏 ∈ (𝑔(𝑡 Nat 𝑢)), 𝑎 ∈ (𝑓(𝑡 Nat 𝑢)𝑔) ↦ (𝑥 ∈ (Base‘𝑡) ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝑢)((1st)‘𝑥))(𝑎𝑥))))) = )
3332opeq2d 4347 . . . 4 ((𝜑 ∧ (𝑡 = 𝐶𝑢 = 𝐷)) → ⟨(comp‘ndx), (𝑣 ∈ ((𝑡 Func 𝑢) × (𝑡 Func 𝑢)), ∈ (𝑡 Func 𝑢) ↦ (1st𝑣) / 𝑓(2nd𝑣) / 𝑔(𝑏 ∈ (𝑔(𝑡 Nat 𝑢)), 𝑎 ∈ (𝑓(𝑡 Nat 𝑢)𝑔) ↦ (𝑥 ∈ (Base‘𝑡) ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝑢)((1st)‘𝑥))(𝑎𝑥)))))⟩ = ⟨(comp‘ndx), ⟩)
349, 13, 33tpeq123d 4227 . . 3 ((𝜑 ∧ (𝑡 = 𝐶𝑢 = 𝐷)) → {⟨(Base‘ndx), (𝑡 Func 𝑢)⟩, ⟨(Hom ‘ndx), (𝑡 Nat 𝑢)⟩, ⟨(comp‘ndx), (𝑣 ∈ ((𝑡 Func 𝑢) × (𝑡 Func 𝑢)), ∈ (𝑡 Func 𝑢) ↦ (1st𝑣) / 𝑓(2nd𝑣) / 𝑔(𝑏 ∈ (𝑔(𝑡 Nat 𝑢)), 𝑎 ∈ (𝑓(𝑡 Nat 𝑢)𝑔) ↦ (𝑥 ∈ (Base‘𝑡) ↦ ((𝑏𝑥)(⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝑢)((1st)‘𝑥))(𝑎𝑥)))))⟩} = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝑁⟩, ⟨(comp‘ndx), ⟩})
35 fucval.c . . 3 (𝜑𝐶 ∈ Cat)
36 fucval.d . . 3 (𝜑𝐷 ∈ Cat)
37 tpex 6855 . . . 4 {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝑁⟩, ⟨(comp‘ndx), ⟩} ∈ V
3837a1i 11 . . 3 (𝜑 → {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝑁⟩, ⟨(comp‘ndx), ⟩} ∈ V)
393, 34, 35, 36, 38ovmpt2d 6686 . 2 (𝜑 → (𝐶 FuncCat 𝐷) = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝑁⟩, ⟨(comp‘ndx), ⟩})
401, 39syl5eq 2656 1 (𝜑𝑄 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝑁⟩, ⟨(comp‘ndx), ⟩})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ⦋csb 3499  {ctp 4129  ⟨cop 4131   ↦ cmpt 4643   × cxp 5036  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  1st c1st 7057  2nd c2nd 7058  ndxcnx 15692  Basecbs 15695  Hom chom 15779  compcco 15780  Catccat 16148   Func cfunc 16337   Nat cnat 16424   FuncCat cfuc 16425 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-fuc 16427 This theorem is referenced by:  fuccofval  16442  fucbas  16443  fuchom  16444  fucpropd  16460  catcfuccl  16582
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