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Theorem prstcnidlem 45852
Description: Lemma for prstcnid 45853 and prstchomval 45859. (Contributed by Zhi Wang, 20-Sep-2024.) (New usage is discouraged.)
Hypotheses
Ref Expression
prstcnid.c (𝜑𝐶 = (ProsetToCat‘𝐾))
prstcnid.k (𝜑𝐾 ∈ Proset )
prstcnid.e 𝐸 = Slot (𝐸‘ndx)
prstcnid.no (𝐸‘ndx) ≠ (comp‘ndx)
Assertion
Ref Expression
prstcnidlem (𝜑 → (𝐸𝐶) = (𝐸‘(𝐾 sSet ⟨(Hom ‘ndx), ((le‘𝐾) × {1o})⟩)))

Proof of Theorem prstcnidlem
StepHypRef Expression
1 prstcnid.c . . . 4 (𝜑𝐶 = (ProsetToCat‘𝐾))
2 prstcnid.k . . . 4 (𝜑𝐾 ∈ Proset )
31, 2prstcval 45851 . . 3 (𝜑𝐶 = ((𝐾 sSet ⟨(Hom ‘ndx), ((le‘𝐾) × {1o})⟩) sSet ⟨(comp‘ndx), ∅⟩))
43fveq2d 6690 . 2 (𝜑 → (𝐸𝐶) = (𝐸‘((𝐾 sSet ⟨(Hom ‘ndx), ((le‘𝐾) × {1o})⟩) sSet ⟨(comp‘ndx), ∅⟩)))
5 prstcnid.e . . 3 𝐸 = Slot (𝐸‘ndx)
6 prstcnid.no . . 3 (𝐸‘ndx) ≠ (comp‘ndx)
75, 6setsnid 16654 . 2 (𝐸‘(𝐾 sSet ⟨(Hom ‘ndx), ((le‘𝐾) × {1o})⟩)) = (𝐸‘((𝐾 sSet ⟨(Hom ‘ndx), ((le‘𝐾) × {1o})⟩) sSet ⟨(comp‘ndx), ∅⟩))
84, 7eqtr4di 2792 1 (𝜑 → (𝐸𝐶) = (𝐸‘(𝐾 sSet ⟨(Hom ‘ndx), ((le‘𝐾) × {1o})⟩)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2114  wne 2935  c0 4221  {csn 4526  cop 4532   × cxp 5533  cfv 6349  (class class class)co 7182  1oc1o 8136  ndxcnx 16595   sSet csts 16596  Slot cslot 16597  lecple 16687  Hom chom 16691  compcco 16692   Proset cproset 17664  ProsetToCatcprstc 45849
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-sep 5177  ax-nul 5184  ax-pr 5306  ax-un 7491
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3402  df-sbc 3686  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4222  df-if 4425  df-sn 4527  df-pr 4529  df-op 4533  df-uni 4807  df-br 5041  df-opab 5103  df-mpt 5121  df-id 5439  df-xp 5541  df-rel 5542  df-cnv 5543  df-co 5544  df-dm 5545  df-res 5547  df-iota 6307  df-fun 6351  df-fv 6357  df-ov 7185  df-oprab 7186  df-mpo 7187  df-slot 16602  df-sets 16605  df-prstc 45850
This theorem is referenced by:  prstcnid  45853  prstchomval  45859
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