MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  setcval Structured version   Visualization version   GIF version

Theorem setcval 17328
Description: Value of the category of sets (in a universe). (Contributed by Mario Carneiro, 3-Jan-2017.)
Hypotheses
Ref Expression
setcval.c 𝐶 = (SetCat‘𝑈)
setcval.u (𝜑𝑈𝑉)
setcval.h (𝜑𝐻 = (𝑥𝑈, 𝑦𝑈 ↦ (𝑦m 𝑥)))
setcval.o (𝜑· = (𝑣 ∈ (𝑈 × 𝑈), 𝑧𝑈 ↦ (𝑔 ∈ (𝑧m (2nd𝑣)), 𝑓 ∈ ((2nd𝑣) ↑m (1st𝑣)) ↦ (𝑔𝑓))))
Assertion
Ref Expression
setcval (𝜑𝐶 = {⟨(Base‘ndx), 𝑈⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})
Distinct variable groups:   𝑓,𝑔,𝑣,𝑥,𝑦,𝑧   𝜑,𝑣,𝑥,𝑦,𝑧   𝑣,𝑈,𝑥,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑓,𝑔)   𝐶(𝑥,𝑦,𝑧,𝑣,𝑓,𝑔)   · (𝑥,𝑦,𝑧,𝑣,𝑓,𝑔)   𝑈(𝑓,𝑔)   𝐻(𝑥,𝑦,𝑧,𝑣,𝑓,𝑔)   𝑉(𝑥,𝑦,𝑧,𝑣,𝑓,𝑔)

Proof of Theorem setcval
Dummy variable 𝑢 is distinct from all other variables.
StepHypRef Expression
1 setcval.c . 2 𝐶 = (SetCat‘𝑈)
2 df-setc 17327 . . 3 SetCat = (𝑢 ∈ V ↦ {⟨(Base‘ndx), 𝑢⟩, ⟨(Hom ‘ndx), (𝑥𝑢, 𝑦𝑢 ↦ (𝑦m 𝑥))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝑢 × 𝑢), 𝑧𝑢 ↦ (𝑔 ∈ (𝑧m (2nd𝑣)), 𝑓 ∈ ((2nd𝑣) ↑m (1st𝑣)) ↦ (𝑔𝑓)))⟩})
3 simpr 488 . . . . 5 ((𝜑𝑢 = 𝑈) → 𝑢 = 𝑈)
43opeq2d 4785 . . . 4 ((𝜑𝑢 = 𝑈) → ⟨(Base‘ndx), 𝑢⟩ = ⟨(Base‘ndx), 𝑈⟩)
5 eqidd 2823 . . . . . . 7 ((𝜑𝑢 = 𝑈) → (𝑦m 𝑥) = (𝑦m 𝑥))
63, 3, 5mpoeq123dv 7213 . . . . . 6 ((𝜑𝑢 = 𝑈) → (𝑥𝑢, 𝑦𝑢 ↦ (𝑦m 𝑥)) = (𝑥𝑈, 𝑦𝑈 ↦ (𝑦m 𝑥)))
7 setcval.h . . . . . . 7 (𝜑𝐻 = (𝑥𝑈, 𝑦𝑈 ↦ (𝑦m 𝑥)))
87adantr 484 . . . . . 6 ((𝜑𝑢 = 𝑈) → 𝐻 = (𝑥𝑈, 𝑦𝑈 ↦ (𝑦m 𝑥)))
96, 8eqtr4d 2860 . . . . 5 ((𝜑𝑢 = 𝑈) → (𝑥𝑢, 𝑦𝑢 ↦ (𝑦m 𝑥)) = 𝐻)
109opeq2d 4785 . . . 4 ((𝜑𝑢 = 𝑈) → ⟨(Hom ‘ndx), (𝑥𝑢, 𝑦𝑢 ↦ (𝑦m 𝑥))⟩ = ⟨(Hom ‘ndx), 𝐻⟩)
113sqxpeqd 5564 . . . . . . 7 ((𝜑𝑢 = 𝑈) → (𝑢 × 𝑢) = (𝑈 × 𝑈))
12 eqidd 2823 . . . . . . 7 ((𝜑𝑢 = 𝑈) → (𝑔 ∈ (𝑧m (2nd𝑣)), 𝑓 ∈ ((2nd𝑣) ↑m (1st𝑣)) ↦ (𝑔𝑓)) = (𝑔 ∈ (𝑧m (2nd𝑣)), 𝑓 ∈ ((2nd𝑣) ↑m (1st𝑣)) ↦ (𝑔𝑓)))
1311, 3, 12mpoeq123dv 7213 . . . . . 6 ((𝜑𝑢 = 𝑈) → (𝑣 ∈ (𝑢 × 𝑢), 𝑧𝑢 ↦ (𝑔 ∈ (𝑧m (2nd𝑣)), 𝑓 ∈ ((2nd𝑣) ↑m (1st𝑣)) ↦ (𝑔𝑓))) = (𝑣 ∈ (𝑈 × 𝑈), 𝑧𝑈 ↦ (𝑔 ∈ (𝑧m (2nd𝑣)), 𝑓 ∈ ((2nd𝑣) ↑m (1st𝑣)) ↦ (𝑔𝑓))))
14 setcval.o . . . . . . 7 (𝜑· = (𝑣 ∈ (𝑈 × 𝑈), 𝑧𝑈 ↦ (𝑔 ∈ (𝑧m (2nd𝑣)), 𝑓 ∈ ((2nd𝑣) ↑m (1st𝑣)) ↦ (𝑔𝑓))))
1514adantr 484 . . . . . 6 ((𝜑𝑢 = 𝑈) → · = (𝑣 ∈ (𝑈 × 𝑈), 𝑧𝑈 ↦ (𝑔 ∈ (𝑧m (2nd𝑣)), 𝑓 ∈ ((2nd𝑣) ↑m (1st𝑣)) ↦ (𝑔𝑓))))
1613, 15eqtr4d 2860 . . . . 5 ((𝜑𝑢 = 𝑈) → (𝑣 ∈ (𝑢 × 𝑢), 𝑧𝑢 ↦ (𝑔 ∈ (𝑧m (2nd𝑣)), 𝑓 ∈ ((2nd𝑣) ↑m (1st𝑣)) ↦ (𝑔𝑓))) = · )
1716opeq2d 4785 . . . 4 ((𝜑𝑢 = 𝑈) → ⟨(comp‘ndx), (𝑣 ∈ (𝑢 × 𝑢), 𝑧𝑢 ↦ (𝑔 ∈ (𝑧m (2nd𝑣)), 𝑓 ∈ ((2nd𝑣) ↑m (1st𝑣)) ↦ (𝑔𝑓)))⟩ = ⟨(comp‘ndx), · ⟩)
184, 10, 17tpeq123d 4658 . . 3 ((𝜑𝑢 = 𝑈) → {⟨(Base‘ndx), 𝑢⟩, ⟨(Hom ‘ndx), (𝑥𝑢, 𝑦𝑢 ↦ (𝑦m 𝑥))⟩, ⟨(comp‘ndx), (𝑣 ∈ (𝑢 × 𝑢), 𝑧𝑢 ↦ (𝑔 ∈ (𝑧m (2nd𝑣)), 𝑓 ∈ ((2nd𝑣) ↑m (1st𝑣)) ↦ (𝑔𝑓)))⟩} = {⟨(Base‘ndx), 𝑈⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})
19 setcval.u . . . 4 (𝜑𝑈𝑉)
2019elexd 3489 . . 3 (𝜑𝑈 ∈ V)
21 tpex 7455 . . . 4 {⟨(Base‘ndx), 𝑈⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩} ∈ V
2221a1i 11 . . 3 (𝜑 → {⟨(Base‘ndx), 𝑈⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩} ∈ V)
232, 18, 20, 22fvmptd2 6758 . 2 (𝜑 → (SetCat‘𝑈) = {⟨(Base‘ndx), 𝑈⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})
241, 23syl5eq 2869 1 (𝜑𝐶 = {⟨(Base‘ndx), 𝑈⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2114  Vcvv 3469  {ctp 4543  cop 4545   × cxp 5530  ccom 5536  cfv 6334  (class class class)co 7140  cmpo 7142  1st c1st 7673  2nd c2nd 7674  m cmap 8393  ndxcnx 16471  Basecbs 16474  Hom chom 16567  compcco 16568  SetCatcsetc 17326
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 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307  ax-un 7446
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 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-iota 6293  df-fun 6336  df-fv 6342  df-oprab 7144  df-mpo 7145  df-setc 17327
This theorem is referenced by:  setcbas  17329  setchomfval  17330  setccofval  17333
  Copyright terms: Public domain W3C validator