Theorem prstcval 50048

Description: Lemma for prstcnidlem 50049 and prstcthin 50058. (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 6834	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
5	4	xpeq1d 5654	. . . . . . 7 ⊢ (𝑘 = 𝐾 → ((le‘𝑘) × {1o}) = ((le‘𝐾) × {1o}))
6	5	opeq2d 4818	. . . . . 6 ⊢ (𝑘 = 𝐾 → ⟨(Hom ‘ndx), ((le‘𝑘) × {1o})⟩ = ⟨(Hom ‘ndx), ((le‘𝐾) × {1o})⟩)
7	3, 6	oveq12d 7381	. . . . 5 ⊢ (𝑘 = 𝐾 → (𝑘 sSet ⟨(Hom ‘ndx), ((le‘𝑘) × {1o})⟩) = (𝐾 sSet ⟨(Hom ‘ndx), ((le‘𝐾) × {1o})⟩))
8	7	oveq1d 7378	. . . 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 50047	. . . 4 ⊢ ProsetToCat = (𝑘 ∈ Proset ↦ ((𝑘 sSet ⟨(Hom ‘ndx), ((le‘𝑘) × {1o})⟩) sSet ⟨(comp‘ndx), ∅⟩))
10		ovex 7396	. . . 4 ⊢ ((𝐾 sSet ⟨(Hom ‘ndx), ((le‘𝐾) × {1o})⟩) sSet ⟨(comp‘ndx), ∅⟩) ∈ V
11	8, 9, 10	fvmpt 6942	. . 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 2775	1 ⊢ (𝜑 → 𝐶 = ((𝐾 sSet ⟨(Hom ‘ndx), ((le‘𝐾) × {1o})⟩) sSet ⟨(comp‘ndx), ∅⟩))

Syntax hints:   → wi 4   = wceq 1547   ∈ wcel 2119  ∅c0 4268  {csn 4562  ⟨cop 4568   × cxp 5623  ‘cfv 6492  (class class class)co 7363  1_oc1o 8395   sSet csts 17131  ndxcnx 17161  lecple 17225  Hom chom 17229  compcco 17230   Proset cproset 18256  ProsetToCatcprstc 50046

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 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-iota 6448  df-fun 6494  df-fv 6500  df-ov 7366  df-prstc 50047

This theorem is referenced by:  prstcnidlem  50049  prstcthin  50058