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Theorem prstcnid 45790
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 16635 . 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 45789 . 2 (𝜑 → (𝐸𝐶) = (𝐸‘(𝐾 sSet ⟨(Hom ‘ndx), ((le‘𝐾) × {1o})⟩)))
83, 7eqtr4id 2792 1 (𝜑 → (𝐸𝐾) = (𝐸𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2113  wne 2934  {csn 4513  cop 4519   × cxp 5517  cfv 6333  (class class class)co 7164  1oc1o 8117  ndxcnx 16576   sSet csts 16577  Slot cslot 16578  lecple 16668  Hom chom 16672  compcco 16673   Proset cproset 17645  ProsetToCatcprstc 45786
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 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-sep 5164  ax-nul 5171  ax-pr 5293  ax-un 7473
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 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3399  df-sbc 3680  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-nul 4210  df-if 4412  df-sn 4514  df-pr 4516  df-op 4520  df-uni 4794  df-br 5028  df-opab 5090  df-mpt 5108  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-res 5531  df-iota 6291  df-fun 6335  df-fv 6341  df-ov 7167  df-oprab 7168  df-mpo 7169  df-slot 16583  df-sets 16586  df-prstc 45787
This theorem is referenced by:  prstcbas  45791  prstcleval  45792  prstcocval  45794
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