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Theorem setcval 17325
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 17324 . . 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 485 . . . . 5 ((𝜑𝑢 = 𝑈) → 𝑢 = 𝑈)
43opeq2d 4802 . . . 4 ((𝜑𝑢 = 𝑈) → ⟨(Base‘ndx), 𝑢⟩ = ⟨(Base‘ndx), 𝑈⟩)
5 eqidd 2819 . . . . . . 7 ((𝜑𝑢 = 𝑈) → (𝑦m 𝑥) = (𝑦m 𝑥))
63, 3, 5mpoeq123dv 7218 . . . . . 6 ((𝜑𝑢 = 𝑈) → (𝑥𝑢, 𝑦𝑢 ↦ (𝑦m 𝑥)) = (𝑥𝑈, 𝑦𝑈 ↦ (𝑦m 𝑥)))
7 setcval.h . . . . . . 7 (𝜑𝐻 = (𝑥𝑈, 𝑦𝑈 ↦ (𝑦m 𝑥)))
87adantr 481 . . . . . 6 ((𝜑𝑢 = 𝑈) → 𝐻 = (𝑥𝑈, 𝑦𝑈 ↦ (𝑦m 𝑥)))
96, 8eqtr4d 2856 . . . . 5 ((𝜑𝑢 = 𝑈) → (𝑥𝑢, 𝑦𝑢 ↦ (𝑦m 𝑥)) = 𝐻)
109opeq2d 4802 . . . 4 ((𝜑𝑢 = 𝑈) → ⟨(Hom ‘ndx), (𝑥𝑢, 𝑦𝑢 ↦ (𝑦m 𝑥))⟩ = ⟨(Hom ‘ndx), 𝐻⟩)
113sqxpeqd 5580 . . . . . . 7 ((𝜑𝑢 = 𝑈) → (𝑢 × 𝑢) = (𝑈 × 𝑈))
12 eqidd 2819 . . . . . . 7 ((𝜑𝑢 = 𝑈) → (𝑔 ∈ (𝑧m (2nd𝑣)), 𝑓 ∈ ((2nd𝑣) ↑m (1st𝑣)) ↦ (𝑔𝑓)) = (𝑔 ∈ (𝑧m (2nd𝑣)), 𝑓 ∈ ((2nd𝑣) ↑m (1st𝑣)) ↦ (𝑔𝑓)))
1311, 3, 12mpoeq123dv 7218 . . . . . 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 481 . . . . . 6 ((𝜑𝑢 = 𝑈) → · = (𝑣 ∈ (𝑈 × 𝑈), 𝑧𝑈 ↦ (𝑔 ∈ (𝑧m (2nd𝑣)), 𝑓 ∈ ((2nd𝑣) ↑m (1st𝑣)) ↦ (𝑔𝑓))))
1613, 15eqtr4d 2856 . . . . 5 ((𝜑𝑢 = 𝑈) → (𝑣 ∈ (𝑢 × 𝑢), 𝑧𝑢 ↦ (𝑔 ∈ (𝑧m (2nd𝑣)), 𝑓 ∈ ((2nd𝑣) ↑m (1st𝑣)) ↦ (𝑔𝑓))) = · )
1716opeq2d 4802 . . . 4 ((𝜑𝑢 = 𝑈) → ⟨(comp‘ndx), (𝑣 ∈ (𝑢 × 𝑢), 𝑧𝑢 ↦ (𝑔 ∈ (𝑧m (2nd𝑣)), 𝑓 ∈ ((2nd𝑣) ↑m (1st𝑣)) ↦ (𝑔𝑓)))⟩ = ⟨(comp‘ndx), · ⟩)
184, 10, 17tpeq123d 4676 . . 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 3512 . . 3 (𝜑𝑈 ∈ V)
21 tpex 7459 . . . 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 6768 . 2 (𝜑 → (SetCat‘𝑈) = {⟨(Base‘ndx), 𝑈⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})
241, 23syl5eq 2865 1 (𝜑𝐶 = {⟨(Base‘ndx), 𝑈⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  Vcvv 3492  {ctp 4561  cop 4563   × cxp 5546  ccom 5552  cfv 6348  (class class class)co 7145  cmpo 7147  1st c1st 7676  2nd c2nd 7677  m cmap 8395  ndxcnx 16468  Basecbs 16471  Hom chom 16564  compcco 16565  SetCatcsetc 17323
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-oprab 7149  df-mpo 7150  df-setc 17324
This theorem is referenced by:  setcbas  17326  setchomfval  17327  setccofval  17330
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