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Theorem fucval 18027
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 18012 . . . 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 770 . . . . . . 7 ((𝜑 ∧ (𝑡 = 𝐶𝑢 = 𝐷)) → 𝑡 = 𝐶)
5 simprr 772 . . . . . . 7 ((𝜑 ∧ (𝑡 = 𝐶𝑢 = 𝐷)) → 𝑢 = 𝐷)
64, 5oveq12d 7466 . . . . . 6 ((𝜑 ∧ (𝑡 = 𝐶𝑢 = 𝐷)) → (𝑡 Func 𝑢) = (𝐶 Func 𝐷))
7 fucval.b . . . . . 6 𝐵 = (𝐶 Func 𝐷)
86, 7eqtr4di 2798 . . . . 5 ((𝜑 ∧ (𝑡 = 𝐶𝑢 = 𝐷)) → (𝑡 Func 𝑢) = 𝐵)
98opeq2d 4904 . . . 4 ((𝜑 ∧ (𝑡 = 𝐶𝑢 = 𝐷)) → ⟨(Base‘ndx), (𝑡 Func 𝑢)⟩ = ⟨(Base‘ndx), 𝐵⟩)
104, 5oveq12d 7466 . . . . . 6 ((𝜑 ∧ (𝑡 = 𝐶𝑢 = 𝐷)) → (𝑡 Nat 𝑢) = (𝐶 Nat 𝐷))
11 fucval.n . . . . . 6 𝑁 = (𝐶 Nat 𝐷)
1210, 11eqtr4di 2798 . . . . 5 ((𝜑 ∧ (𝑡 = 𝐶𝑢 = 𝐷)) → (𝑡 Nat 𝑢) = 𝑁)
1312opeq2d 4904 . . . 4 ((𝜑 ∧ (𝑡 = 𝐶𝑢 = 𝐷)) → ⟨(Hom ‘ndx), (𝑡 Nat 𝑢)⟩ = ⟨(Hom ‘ndx), 𝑁⟩)
148sqxpeqd 5732 . . . . . . 7 ((𝜑 ∧ (𝑡 = 𝐶𝑢 = 𝐷)) → ((𝑡 Func 𝑢) × (𝑡 Func 𝑢)) = (𝐵 × 𝐵))
1512oveqd 7465 . . . . . . . . . 10 ((𝜑 ∧ (𝑡 = 𝐶𝑢 = 𝐷)) → (𝑔(𝑡 Nat 𝑢)) = (𝑔𝑁))
1612oveqd 7465 . . . . . . . . . 10 ((𝜑 ∧ (𝑡 = 𝐶𝑢 = 𝐷)) → (𝑓(𝑡 Nat 𝑢)𝑔) = (𝑓𝑁𝑔))
174fveq2d 6924 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑡 = 𝐶𝑢 = 𝐷)) → (Base‘𝑡) = (Base‘𝐶))
18 fucval.a . . . . . . . . . . . 12 𝐴 = (Base‘𝐶)
1917, 18eqtr4di 2798 . . . . . . . . . . 11 ((𝜑 ∧ (𝑡 = 𝐶𝑢 = 𝐷)) → (Base‘𝑡) = 𝐴)
205fveq2d 6924 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑡 = 𝐶𝑢 = 𝐷)) → (comp‘𝑢) = (comp‘𝐷))
21 fucval.o . . . . . . . . . . . . . 14 · = (comp‘𝐷)
2220, 21eqtr4di 2798 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑡 = 𝐶𝑢 = 𝐷)) → (comp‘𝑢) = · )
2322oveqd 7465 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑡 = 𝐶𝑢 = 𝐷)) → (⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩(comp‘𝑢)((1st)‘𝑥)) = (⟨((1st𝑓)‘𝑥), ((1st𝑔)‘𝑥)⟩ · ((1st)‘𝑥)))
2423oveqd 7465 . . . . . . . . . . 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 5257 . . . . . . . . . 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 7525 . . . . . . . . 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 3928 . . . . . . . 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 3928 . . . . . . 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 7525 . . . . . 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 2783 . . . . 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 4904 . . . 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 4773 . . 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 7781 . . . 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 7602 . 2 (𝜑 → (𝐶 FuncCat 𝐷) = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝑁⟩, ⟨(comp‘ndx), ⟩})
401, 39eqtrid 2792 1 (𝜑𝑄 = {⟨(Base‘ndx), 𝐵⟩, ⟨(Hom ‘ndx), 𝑁⟩, ⟨(comp‘ndx), ⟩})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2108  Vcvv 3488  csb 3921  {ctp 4652  cop 4654  cmpt 5249   × cxp 5698  cfv 6573  (class class class)co 7448  cmpo 7450  1st c1st 8028  2nd c2nd 8029  ndxcnx 17240  Basecbs 17258  Hom chom 17322  compcco 17323  Catccat 17722   Func cfunc 17918   Nat cnat 18009   FuncCat cfuc 18010
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-iota 6525  df-fun 6575  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-fuc 18012
This theorem is referenced by:  fuccofval  18028  fucbas  18029  fuchom  18030  fuchomOLD  18031  fucpropd  18047  catcfuccl  18186  catcfucclOLD  18187
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