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Theorem prstcnid 47176
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 17089 . 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 47175 . 2 (𝜑 → (𝐸𝐶) = (𝐸‘(𝐾 sSet ⟨(Hom ‘ndx), ((le‘𝐾) × {1o})⟩)))
83, 7eqtr4id 2791 1 (𝜑 → (𝐸𝐾) = (𝐸𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  wne 2940  {csn 4590  cop 4596   × cxp 5635  cfv 6500  (class class class)co 7361  1oc1o 8409   sSet csts 17043  Slot cslot 17061  ndxcnx 17073  lecple 17148  Hom chom 17152  compcco 17153   Proset cproset 18190  ProsetToCatcprstc 47172
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 2703  ax-sep 5260  ax-nul 5267  ax-pr 5388  ax-un 7676
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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-sbc 3744  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-res 5649  df-iota 6452  df-fun 6502  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-sets 17044  df-slot 17062  df-prstc 47173
This theorem is referenced by:  prstcbas  47177  prstcleval  47178  prstclevalOLD  47179  prstcocval  47181  prstcocvalOLD  47182
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