Theorem prstcnid 45790

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 16635	. 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 45789	. 2 ⊢ (𝜑 → (𝐸‘𝐶) = (𝐸‘(𝐾 sSet ⟨(Hom ‘ndx), ((le‘𝐾) × {1o})⟩)))
8	3, 7	eqtr4id 2792	1 ⊢ (𝜑 → (𝐸‘𝐾) = (𝐸‘𝐶))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2113   ≠ wne 2934  {csn 4513  ⟨cop 4519   × cxp 5517  ‘cfv 6333  (class class class)co 7164  1_oc1o 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