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Theorem prstcnid 49155
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 17245 . 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 49154 . 2 (𝜑 → (𝐸𝐶) = (𝐸‘(𝐾 sSet ⟨(Hom ‘ndx), ((le‘𝐾) × {1o})⟩)))
83, 7eqtr4id 2796 1 (𝜑 → (𝐸𝐾) = (𝐸𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  wne 2940  {csn 4626  cop 4632   × cxp 5683  cfv 6561  (class class class)co 7431  1oc1o 8499   sSet csts 17200  Slot cslot 17218  ndxcnx 17230  lecple 17304  Hom chom 17308  compcco 17309   Proset cproset 18338  ProsetToCatcprstc 49151
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-res 5697  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-sets 17201  df-slot 17219  df-prstc 49152
This theorem is referenced by:  prstcbas  49156  prstcleval  49157  prstclevalOLD  49158  prstcocval  49160  prstcocvalOLD  49161
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