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Theorem prstcval 47524
Description: Lemma for prstcnidlem 47525 and prstcthin 47536. (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 6881 . . . . . . . 8 (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
54xpeq1d 5701 . . . . . . 7 (𝑘 = 𝐾 → ((le‘𝑘) × {1o}) = ((le‘𝐾) × {1o}))
65opeq2d 4876 . . . . . 6 (𝑘 = 𝐾 → ⟨(Hom ‘ndx), ((le‘𝑘) × {1o})⟩ = ⟨(Hom ‘ndx), ((le‘𝐾) × {1o})⟩)
73, 6oveq12d 7414 . . . . 5 (𝑘 = 𝐾 → (𝑘 sSet ⟨(Hom ‘ndx), ((le‘𝑘) × {1o})⟩) = (𝐾 sSet ⟨(Hom ‘ndx), ((le‘𝐾) × {1o})⟩))
87oveq1d 7411 . . . 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 47523 . . . 4 ProsetToCat = (𝑘 ∈ Proset ↦ ((𝑘 sSet ⟨(Hom ‘ndx), ((le‘𝑘) × {1o})⟩) sSet ⟨(comp‘ndx), ∅⟩))
10 ovex 7429 . . . 4 ((𝐾 sSet ⟨(Hom ‘ndx), ((le‘𝐾) × {1o})⟩) sSet ⟨(comp‘ndx), ∅⟩) ∈ V
118, 9, 10fvmpt 6987 . . 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 2773 1 (𝜑𝐶 = ((𝐾 sSet ⟨(Hom ‘ndx), ((le‘𝐾) × {1o})⟩) sSet ⟨(comp‘ndx), ∅⟩))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  c0 4320  {csn 4624  cop 4630   × cxp 5670  cfv 6535  (class class class)co 7396  1oc1o 8446   sSet csts 17083  ndxcnx 17113  lecple 17191  Hom chom 17195  compcco 17196   Proset cproset 18233  ProsetToCatcprstc 47522
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5295  ax-nul 5302  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4905  df-br 5145  df-opab 5207  df-mpt 5228  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-iota 6487  df-fun 6537  df-fv 6543  df-ov 7399  df-prstc 47523
This theorem is referenced by:  prstcnidlem  47525  prstcthin  47536
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