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Theorem prstcval 46345
Description: Lemma for prstcnidlem 46346 and prstcthin 46357. (Contributed by Zhi Wang, 20-Sep-2024.) (New usage is discouraged.)
Hypotheses
Ref Expression
prstcnid.c (𝜑𝐶 = (ProsetToCat‘𝐾))
prstcnid.k (𝜑𝐾 ∈ Proset )
Assertion
Ref Expression
prstcval (𝜑𝐶 = ((𝐾 sSet ⟨(Hom ‘ndx), ((le‘𝐾) × {1o})⟩) sSet ⟨(comp‘ndx), ∅⟩))

Proof of Theorem prstcval
Dummy variable 𝑘 is distinct from all other variables.
StepHypRef Expression
1 prstcnid.c . 2 (𝜑𝐶 = (ProsetToCat‘𝐾))
2 prstcnid.k . . 3 (𝜑𝐾 ∈ Proset )
3 id 22 . . . . . 6 (𝑘 = 𝐾𝑘 = 𝐾)
4 fveq2 6774 . . . . . . . 8 (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
54xpeq1d 5618 . . . . . . 7 (𝑘 = 𝐾 → ((le‘𝑘) × {1o}) = ((le‘𝐾) × {1o}))
65opeq2d 4811 . . . . . 6 (𝑘 = 𝐾 → ⟨(Hom ‘ndx), ((le‘𝑘) × {1o})⟩ = ⟨(Hom ‘ndx), ((le‘𝐾) × {1o})⟩)
73, 6oveq12d 7293 . . . . 5 (𝑘 = 𝐾 → (𝑘 sSet ⟨(Hom ‘ndx), ((le‘𝑘) × {1o})⟩) = (𝐾 sSet ⟨(Hom ‘ndx), ((le‘𝐾) × {1o})⟩))
87oveq1d 7290 . . . 4 (𝑘 = 𝐾 → ((𝑘 sSet ⟨(Hom ‘ndx), ((le‘𝑘) × {1o})⟩) sSet ⟨(comp‘ndx), ∅⟩) = ((𝐾 sSet ⟨(Hom ‘ndx), ((le‘𝐾) × {1o})⟩) sSet ⟨(comp‘ndx), ∅⟩))
9 df-prstc 46344 . . . 4 ProsetToCat = (𝑘 ∈ Proset ↦ ((𝑘 sSet ⟨(Hom ‘ndx), ((le‘𝑘) × {1o})⟩) sSet ⟨(comp‘ndx), ∅⟩))
10 ovex 7308 . . . 4 ((𝐾 sSet ⟨(Hom ‘ndx), ((le‘𝐾) × {1o})⟩) sSet ⟨(comp‘ndx), ∅⟩) ∈ V
118, 9, 10fvmpt 6875 . . 3 (𝐾 ∈ Proset → (ProsetToCat‘𝐾) = ((𝐾 sSet ⟨(Hom ‘ndx), ((le‘𝐾) × {1o})⟩) sSet ⟨(comp‘ndx), ∅⟩))
122, 11syl 17 . 2 (𝜑 → (ProsetToCat‘𝐾) = ((𝐾 sSet ⟨(Hom ‘ndx), ((le‘𝐾) × {1o})⟩) sSet ⟨(comp‘ndx), ∅⟩))
131, 12eqtrd 2778 1 (𝜑𝐶 = ((𝐾 sSet ⟨(Hom ‘ndx), ((le‘𝐾) × {1o})⟩) sSet ⟨(comp‘ndx), ∅⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wcel 2106  c0 4256  {csn 4561  cop 4567   × cxp 5587  cfv 6433  (class class class)co 7275  1oc1o 8290   sSet csts 16864  ndxcnx 16894  lecple 16969  Hom chom 16973  compcco 16974   Proset cproset 18011  ProsetToCatcprstc 46343
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-iota 6391  df-fun 6435  df-fv 6441  df-ov 7278  df-prstc 46344
This theorem is referenced by:  prstcnidlem  46346  prstcthin  46357
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