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Theorem prstcnid 47995
Description: Components other than Hom and comp are unchanged. (Contributed by Zhi Wang, 20-Sep-2024.)
Hypotheses
Ref Expression
prstcnid.c (𝜑𝐶 = (ProsetToCat‘𝐾))
prstcnid.k (𝜑𝐾 ∈ Proset )
prstcnid.e 𝐸 = Slot (𝐸‘ndx)
prstcnid.no (𝐸‘ndx) ≠ (comp‘ndx)
prstcnid.nh (𝐸‘ndx) ≠ (Hom ‘ndx)
Assertion
Ref Expression
prstcnid (𝜑 → (𝐸𝐾) = (𝐸𝐶))

Proof of Theorem prstcnid
StepHypRef Expression
1 prstcnid.e . . 3 𝐸 = Slot (𝐸‘ndx)
2 prstcnid.nh . . 3 (𝐸‘ndx) ≠ (Hom ‘ndx)
31, 2setsnid 17169 . 2 (𝐸𝐾) = (𝐸‘(𝐾 sSet ⟨(Hom ‘ndx), ((le‘𝐾) × {1o})⟩))
4 prstcnid.c . . 3 (𝜑𝐶 = (ProsetToCat‘𝐾))
5 prstcnid.k . . 3 (𝜑𝐾 ∈ Proset )
6 prstcnid.no . . 3 (𝐸‘ndx) ≠ (comp‘ndx)
74, 5, 1, 6prstcnidlem 47994 . 2 (𝜑 → (𝐸𝐶) = (𝐸‘(𝐾 sSet ⟨(Hom ‘ndx), ((le‘𝐾) × {1o})⟩)))
83, 7eqtr4id 2786 1 (𝜑 → (𝐸𝐾) = (𝐸𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1534  wcel 2099  wne 2935  {csn 4624  cop 4630   × cxp 5670  cfv 6542  (class class class)co 7414  1oc1o 8473   sSet csts 17123  Slot cslot 17141  ndxcnx 17153  lecple 17231  Hom chom 17235  compcco 17236   Proset cproset 18276  ProsetToCatcprstc 47991
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-sbc 3775  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-res 5684  df-iota 6494  df-fun 6544  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419  df-sets 17124  df-slot 17142  df-prstc 47992
This theorem is referenced by:  prstcbas  47996  prstcleval  47997  prstclevalOLD  47998  prstcocval  48000  prstcocvalOLD  48001
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