Theorem prstcval 49429

Description: Lemma for prstcnidlem 49430 and prstcthin 49439. (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 6865	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
5	4	xpeq1d 5675	. . . . . . 7 ⊢ (𝑘 = 𝐾 → ((le‘𝑘) × {1o}) = ((le‘𝐾) × {1o}))
6	5	opeq2d 4852	. . . . . 6 ⊢ (𝑘 = 𝐾 → ⟨(Hom ‘ndx), ((le‘𝑘) × {1o})⟩ = ⟨(Hom ‘ndx), ((le‘𝐾) × {1o})⟩)
7	3, 6	oveq12d 7412	. . . . 5 ⊢ (𝑘 = 𝐾 → (𝑘 sSet ⟨(Hom ‘ndx), ((le‘𝑘) × {1o})⟩) = (𝐾 sSet ⟨(Hom ‘ndx), ((le‘𝐾) × {1o})⟩))
8	7	oveq1d 7409	. . . 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 49428	. . . 4 ⊢ ProsetToCat = (𝑘 ∈ Proset ↦ ((𝑘 sSet ⟨(Hom ‘ndx), ((le‘𝑘) × {1o})⟩) sSet ⟨(comp‘ndx), ∅⟩))
10		ovex 7427	. . . 4 ⊢ ((𝐾 sSet ⟨(Hom ‘ndx), ((le‘𝐾) × {1o})⟩) sSet ⟨(comp‘ndx), ∅⟩) ∈ V
11	8, 9, 10	fvmpt 6975	. . 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 2765	1 ⊢ (𝜑 → 𝐶 = ((𝐾 sSet ⟨(Hom ‘ndx), ((le‘𝐾) × {1o})⟩) sSet ⟨(comp‘ndx), ∅⟩))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  ∅c0 4304  {csn 4597  ⟨cop 4603   × cxp 5644  ‘cfv 6519  (class class class)co 7394  1_oc1o 8436   sSet csts 17139  ndxcnx 17169  lecple 17233  Hom chom 17237  compcco 17238   Proset cproset 18259  ProsetToCatcprstc 49427

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5259  ax-nul 5269  ax-pr 5395

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2880  df-ne 2928  df-ral 3047  df-rex 3056  df-rab 3412  df-v 3457  df-dif 3925  df-un 3927  df-ss 3939  df-nul 4305  df-if 4497  df-sn 4598  df-pr 4600  df-op 4604  df-uni 4880  df-br 5116  df-opab 5178  df-mpt 5197  df-id 5541  df-xp 5652  df-rel 5653  df-cnv 5654  df-co 5655  df-dm 5656  df-iota 6472  df-fun 6521  df-fv 6527  df-ov 7397  df-prstc 49428

This theorem is referenced by:  prstcnidlem  49430  prstcthin  49439