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Theorem setcval 16934
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 16933 . . . 4 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)))⟩})
32a1i 11 . . 3 (𝜑 → 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)))⟩}))
4 simpr 471 . . . . 5 ((𝜑𝑢 = 𝑈) → 𝑢 = 𝑈)
54opeq2d 4546 . . . 4 ((𝜑𝑢 = 𝑈) → ⟨(Base‘ndx), 𝑢⟩ = ⟨(Base‘ndx), 𝑈⟩)
6 eqidd 2772 . . . . . . 7 ((𝜑𝑢 = 𝑈) → (𝑦𝑚 𝑥) = (𝑦𝑚 𝑥))
74, 4, 6mpt2eq123dv 6864 . . . . . 6 ((𝜑𝑢 = 𝑈) → (𝑥𝑢, 𝑦𝑢 ↦ (𝑦𝑚 𝑥)) = (𝑥𝑈, 𝑦𝑈 ↦ (𝑦𝑚 𝑥)))
8 setcval.h . . . . . . 7 (𝜑𝐻 = (𝑥𝑈, 𝑦𝑈 ↦ (𝑦𝑚 𝑥)))
98adantr 466 . . . . . 6 ((𝜑𝑢 = 𝑈) → 𝐻 = (𝑥𝑈, 𝑦𝑈 ↦ (𝑦𝑚 𝑥)))
107, 9eqtr4d 2808 . . . . 5 ((𝜑𝑢 = 𝑈) → (𝑥𝑢, 𝑦𝑢 ↦ (𝑦𝑚 𝑥)) = 𝐻)
1110opeq2d 4546 . . . 4 ((𝜑𝑢 = 𝑈) → ⟨(Hom ‘ndx), (𝑥𝑢, 𝑦𝑢 ↦ (𝑦𝑚 𝑥))⟩ = ⟨(Hom ‘ndx), 𝐻⟩)
124sqxpeqd 5281 . . . . . . 7 ((𝜑𝑢 = 𝑈) → (𝑢 × 𝑢) = (𝑈 × 𝑈))
13 eqidd 2772 . . . . . . 7 ((𝜑𝑢 = 𝑈) → (𝑔 ∈ (𝑧𝑚 (2nd𝑣)), 𝑓 ∈ ((2nd𝑣) ↑𝑚 (1st𝑣)) ↦ (𝑔𝑓)) = (𝑔 ∈ (𝑧𝑚 (2nd𝑣)), 𝑓 ∈ ((2nd𝑣) ↑𝑚 (1st𝑣)) ↦ (𝑔𝑓)))
1412, 4, 13mpt2eq123dv 6864 . . . . . 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))))
15 setcval.o . . . . . . 7 (𝜑· = (𝑣 ∈ (𝑈 × 𝑈), 𝑧𝑈 ↦ (𝑔 ∈ (𝑧𝑚 (2nd𝑣)), 𝑓 ∈ ((2nd𝑣) ↑𝑚 (1st𝑣)) ↦ (𝑔𝑓))))
1615adantr 466 . . . . . 6 ((𝜑𝑢 = 𝑈) → · = (𝑣 ∈ (𝑈 × 𝑈), 𝑧𝑈 ↦ (𝑔 ∈ (𝑧𝑚 (2nd𝑣)), 𝑓 ∈ ((2nd𝑣) ↑𝑚 (1st𝑣)) ↦ (𝑔𝑓))))
1714, 16eqtr4d 2808 . . . . 5 ((𝜑𝑢 = 𝑈) → (𝑣 ∈ (𝑢 × 𝑢), 𝑧𝑢 ↦ (𝑔 ∈ (𝑧𝑚 (2nd𝑣)), 𝑓 ∈ ((2nd𝑣) ↑𝑚 (1st𝑣)) ↦ (𝑔𝑓))) = · )
1817opeq2d 4546 . . . 4 ((𝜑𝑢 = 𝑈) → ⟨(comp‘ndx), (𝑣 ∈ (𝑢 × 𝑢), 𝑧𝑢 ↦ (𝑔 ∈ (𝑧𝑚 (2nd𝑣)), 𝑓 ∈ ((2nd𝑣) ↑𝑚 (1st𝑣)) ↦ (𝑔𝑓)))⟩ = ⟨(comp‘ndx), · ⟩)
195, 11, 18tpeq123d 4419 . . 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), · ⟩})
20 setcval.u . . . 4 (𝜑𝑈𝑉)
21 elex 3364 . . . 4 (𝑈𝑉𝑈 ∈ V)
2220, 21syl 17 . . 3 (𝜑𝑈 ∈ V)
23 tpex 7104 . . . 4 {⟨(Base‘ndx), 𝑈⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩} ∈ V
2423a1i 11 . . 3 (𝜑 → {⟨(Base‘ndx), 𝑈⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩} ∈ V)
253, 19, 22, 24fvmptd 6430 . 2 (𝜑 → (SetCat‘𝑈) = {⟨(Base‘ndx), 𝑈⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})
261, 25syl5eq 2817 1 (𝜑𝐶 = {⟨(Base‘ndx), 𝑈⟩, ⟨(Hom ‘ndx), 𝐻⟩, ⟨(comp‘ndx), · ⟩})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  Vcvv 3351  {ctp 4320  cop 4322  cmpt 4863   × cxp 5247  ccom 5253  cfv 6031  (class class class)co 6793  cmpt2 6795  1st c1st 7313  2nd c2nd 7314  𝑚 cmap 8009  ndxcnx 16061  Basecbs 16064  Hom chom 16160  compcco 16161  SetCatcsetc 16932
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-iota 5994  df-fun 6033  df-fv 6039  df-oprab 6797  df-mpt2 6798  df-setc 16933
This theorem is referenced by:  setcbas  16935  setchomfval  16936  setccofval  16939
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