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Theorem fucval 18006
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 17992 . . . 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 771 . . . . . . 7 ((𝜑 ∧ (𝑡 = 𝐶𝑢 = 𝐷)) → 𝑡 = 𝐶)
5 simprr 773 . . . . . . 7 ((𝜑 ∧ (𝑡 = 𝐶𝑢 = 𝐷)) → 𝑢 = 𝐷)
64, 5oveq12d 7449 . . . . . 6 ((𝜑 ∧ (𝑡 = 𝐶𝑢 = 𝐷)) → (𝑡 Func 𝑢) = (𝐶 Func 𝐷))
7 fucval.b . . . . . 6 𝐵 = (𝐶 Func 𝐷)
86, 7eqtr4di 2795 . . . . 5 ((𝜑 ∧ (𝑡 = 𝐶𝑢 = 𝐷)) → (𝑡 Func 𝑢) = 𝐵)
98opeq2d 4880 . . . 4 ((𝜑 ∧ (𝑡 = 𝐶𝑢 = 𝐷)) → ⟨(Base‘ndx), (𝑡 Func 𝑢)⟩ = ⟨(Base‘ndx), 𝐵⟩)
104, 5oveq12d 7449 . . . . . 6 ((𝜑 ∧ (𝑡 = 𝐶𝑢 = 𝐷)) → (𝑡 Nat 𝑢) = (𝐶 Nat 𝐷))
11 fucval.n . . . . . 6 𝑁 = (𝐶 Nat 𝐷)
1210, 11eqtr4di 2795 . . . . 5 ((𝜑 ∧ (𝑡 = 𝐶𝑢 = 𝐷)) → (𝑡 Nat 𝑢) = 𝑁)
1312opeq2d 4880 . . . 4 ((𝜑 ∧ (𝑡 = 𝐶𝑢 = 𝐷)) → ⟨(Hom ‘ndx), (𝑡 Nat 𝑢)⟩ = ⟨(Hom ‘ndx), 𝑁⟩)
148sqxpeqd 5717 . . . . . . 7 ((𝜑 ∧ (𝑡 = 𝐶𝑢 = 𝐷)) → ((𝑡 Func 𝑢) × (𝑡 Func 𝑢)) = (𝐵 × 𝐵))
1512oveqd 7448 . . . . . . . . . 10 ((𝜑 ∧ (𝑡 = 𝐶𝑢 = 𝐷)) → (𝑔(𝑡 Nat 𝑢)) = (𝑔𝑁))
1612oveqd 7448 . . . . . . . . . 10 ((𝜑 ∧ (𝑡 = 𝐶𝑢 = 𝐷)) → (𝑓(𝑡 Nat 𝑢)𝑔) = (𝑓𝑁𝑔))
174fveq2d 6910 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑡 = 𝐶𝑢 = 𝐷)) → (Base‘𝑡) = (Base‘𝐶))
18 fucval.a . . . . . . . . . . . 12 𝐴 = (Base‘𝐶)
1917, 18eqtr4di 2795 . . . . . . . . . . 11 ((𝜑 ∧ (𝑡 = 𝐶𝑢 = 𝐷)) → (Base‘𝑡) = 𝐴)
205fveq2d 6910 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑡 = 𝐶𝑢 = 𝐷)) → (comp‘𝑢) = (comp‘𝐷))
21 fucval.o . . . . . . . . . . . . . 14 · = (comp‘𝐷)
2220, 21eqtr4di 2795 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑡 = 𝐶𝑢 = 𝐷)) → (comp‘𝑢) = · )
2322oveqd 7448 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑡 = 𝐶𝑢 = 𝐷)) → (⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝑢)((1st)‘𝑥)) = (⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩ · ((1st)‘𝑥)))
2423oveqd 7448 . . . . . . . . . . 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 5233 . . . . . . . . . 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, 25mpoeq123dv 7508 . . . . . . . . 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 3906 . . . . . . . 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 3906 . . . . . . 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, 28mpoeq123dv 7508 . . . . . 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 2780 . . . . 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 4880 . . . 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 4748 . . 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 7766 . . . 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, 38ovmpod 7585 . 2 (𝜑 → (𝐶 FuncCat 𝐷) = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝑁⟩, ⟨(comp‘ndx), ⟩})
401, 39eqtrid 2789 1 (𝜑𝑄 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝑁⟩, ⟨(comp‘ndx), ⟩})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  Vcvv 3480  csb 3899  {ctp 4630  cop 4632  cmpt 5225   × cxp 5683  cfv 6561  (class class class)co 7431  cmpo 7433  1st c1st 8012  2nd c2nd 8013  ndxcnx 17230  Basecbs 17247  Hom chom 17308  compcco 17309  Catccat 17707   Func cfunc 17899   Nat cnat 17989   FuncCat cfuc 17990
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-fuc 17992
This theorem is referenced by:  fuccofval  18007  fucbas  18008  fuchom  18009  fucpropd  18025  catcfuccl  18163
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