Theorem prstcnid 48063

Description: Components other than Hom and comp are unchanged. (Contributed by Zhi Wang, 20-Sep-2024.)

Hypotheses

Ref	Expression
prstcnid.c	⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾))
prstcnid.k	⊢ (𝜑 → 𝐾 ∈ Proset )
prstcnid.e	⊢ 𝐸 = Slot (𝐸‘ndx)
prstcnid.no	⊢ (𝐸‘ndx) ≠ (comp‘ndx)
prstcnid.nh	⊢ (𝐸‘ndx) ≠ (Hom ‘ndx)

Assertion

Ref	Expression
prstcnid	⊢ (𝜑 → (𝐸‘𝐾) = (𝐸‘𝐶))

Proof of Theorem prstcnid

Step	Hyp	Ref	Expression
1		prstcnid.e	. . 3 ⊢ 𝐸 = Slot (𝐸‘ndx)
2		prstcnid.nh	. . 3 ⊢ (𝐸‘ndx) ≠ (Hom ‘ndx)
3	1, 2	setsnid 17172	. 2 ⊢ (𝐸‘𝐾) = (𝐸‘(𝐾 sSet ⟨(Hom ‘ndx), ((le‘𝐾) × {1o})⟩))
4		prstcnid.c	. . 3 ⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾))
5		prstcnid.k	. . 3 ⊢ (𝜑 → 𝐾 ∈ Proset )
6		prstcnid.no	. . 3 ⊢ (𝐸‘ndx) ≠ (comp‘ndx)
7	4, 5, 1, 6	prstcnidlem 48062	. 2 ⊢ (𝜑 → (𝐸‘𝐶) = (𝐸‘(𝐾 sSet ⟨(Hom ‘ndx), ((le‘𝐾) × {1o})⟩)))
8	3, 7	eqtr4id 2787	1 ⊢ (𝜑 → (𝐸‘𝐾) = (𝐸‘𝐶))

Syntax hints:   → wi 4   = wceq 1534   ∈ wcel 2099   ≠ wne 2936  {csn 4625  ⟨cop 4631   × cxp 5671  ‘cfv 6543  (class class class)co 7415  1_oc1o 8474   sSet csts 17126  Slot cslot 17144  ndxcnx 17156  lecple 17234  Hom chom 17238  compcco 17239   Proset cproset 18279  ProsetToCatcprstc 48059

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 2167  ax-ext 2699  ax-sep 5294  ax-nul 5301  ax-pr 5424  ax-un 7735

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 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-sbc 3776  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4526  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-res 5685  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7418  df-oprab 7419  df-mpo 7420  df-sets 17127  df-slot 17145  df-prstc 48060

This theorem is referenced by:  prstcbas  48064  prstcleval  48065  prstclevalOLD  48066  prstcocval  48068  prstcocvalOLD  48069

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