Theorem prstcval 46345

Description: Lemma for prstcnidlem 46346 and prstcthin 46357. (Contributed by Zhi Wang, 20-Sep-2024.) (New usage is discouraged.)

Hypotheses

Ref	Expression
prstcnid.c	⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾))
prstcnid.k	⊢ (𝜑 → 𝐾 ∈ Proset )

Assertion

Ref	Expression
prstcval	⊢ (𝜑 → 𝐶 = ((𝐾 sSet ⟨(Hom ‘ndx), ((le‘𝐾) × {1o})⟩) sSet ⟨(comp‘ndx), ∅⟩))

Proof of Theorem prstcval

Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		prstcnid.c	. 2 ⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾))
2		prstcnid.k	. . 3 ⊢ (𝜑 → 𝐾 ∈ Proset )
3		id 22	. . . . . 6 ⊢ (𝑘 = 𝐾 → 𝑘 = 𝐾)
4		fveq2 6774	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
5	4	xpeq1d 5618	. . . . . . 7 ⊢ (𝑘 = 𝐾 → ((le‘𝑘) × {1o}) = ((le‘𝐾) × {1o}))
6	5	opeq2d 4811	. . . . . 6 ⊢ (𝑘 = 𝐾 → ⟨(Hom ‘ndx), ((le‘𝑘) × {1o})⟩ = ⟨(Hom ‘ndx), ((le‘𝐾) × {1o})⟩)
7	3, 6	oveq12d 7293	. . . . 5 ⊢ (𝑘 = 𝐾 → (𝑘 sSet ⟨(Hom ‘ndx), ((le‘𝑘) × {1o})⟩) = (𝐾 sSet ⟨(Hom ‘ndx), ((le‘𝐾) × {1o})⟩))
8	7	oveq1d 7290	. . . 4 ⊢ (𝑘 = 𝐾 → ((𝑘 sSet ⟨(Hom ‘ndx), ((le‘𝑘) × {1o})⟩) sSet ⟨(comp‘ndx), ∅⟩) = ((𝐾 sSet ⟨(Hom ‘ndx), ((le‘𝐾) × {1o})⟩) sSet ⟨(comp‘ndx), ∅⟩))
9		df-prstc 46344	. . . 4 ⊢ ProsetToCat = (𝑘 ∈ Proset ↦ ((𝑘 sSet ⟨(Hom ‘ndx), ((le‘𝑘) × {1o})⟩) sSet ⟨(comp‘ndx), ∅⟩))
10		ovex 7308	. . . 4 ⊢ ((𝐾 sSet ⟨(Hom ‘ndx), ((le‘𝐾) × {1o})⟩) sSet ⟨(comp‘ndx), ∅⟩) ∈ V
11	8, 9, 10	fvmpt 6875	. . 3 ⊢ (𝐾 ∈ Proset → (ProsetToCat‘𝐾) = ((𝐾 sSet ⟨(Hom ‘ndx), ((le‘𝐾) × {1o})⟩) sSet ⟨(comp‘ndx), ∅⟩))
12	2, 11	syl 17	. 2 ⊢ (𝜑 → (ProsetToCat‘𝐾) = ((𝐾 sSet ⟨(Hom ‘ndx), ((le‘𝐾) × {1o})⟩) sSet ⟨(comp‘ndx), ∅⟩))
13	1, 12	eqtrd 2778	1 ⊢ (𝜑 → 𝐶 = ((𝐾 sSet ⟨(Hom ‘ndx), ((le‘𝐾) × {1o})⟩) sSet ⟨(comp‘ndx), ∅⟩))

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2106  ∅c0 4256  {csn 4561  ⟨cop 4567   × cxp 5587  ‘cfv 6433  (class class class)co 7275  1_oc1o 8290   sSet csts 16864  ndxcnx 16894  lecple 16969  Hom chom 16973  compcco 16974   Proset cproset 18011  ProsetToCatcprstc 46343

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-iota 6391  df-fun 6435  df-fv 6441  df-ov 7278  df-prstc 46344

This theorem is referenced by:  prstcnidlem  46346  prstcthin  46357