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Mirrors > Home > MPE Home > Th. List > Mathboxes > prstcval | Structured version Visualization version GIF version |
Description: Lemma for prstcnidlem 47685 and prstcthin 47696. (Contributed by Zhi Wang, 20-Sep-2024.) (New usage is discouraged.) |
Ref | Expression |
---|---|
prstcnid.c | ⊢ (𝜑 → 𝐶 = (ProsetToCat‘𝐾)) |
prstcnid.k | ⊢ (𝜑 → 𝐾 ∈ Proset ) |
Ref | Expression |
---|---|
prstcval | ⊢ (𝜑 → 𝐶 = ((𝐾 sSet ⟨(Hom ‘ndx), ((le‘𝐾) × {1o})⟩) sSet ⟨(comp‘ndx), ∅⟩)) |
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 6892 | . . . . . . . 8 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾)) | |
5 | 4 | xpeq1d 5706 | . . . . . . 7 ⊢ (𝑘 = 𝐾 → ((le‘𝑘) × {1o}) = ((le‘𝐾) × {1o})) |
6 | 5 | opeq2d 4881 | . . . . . 6 ⊢ (𝑘 = 𝐾 → ⟨(Hom ‘ndx), ((le‘𝑘) × {1o})⟩ = ⟨(Hom ‘ndx), ((le‘𝐾) × {1o})⟩) |
7 | 3, 6 | oveq12d 7427 | . . . . 5 ⊢ (𝑘 = 𝐾 → (𝑘 sSet ⟨(Hom ‘ndx), ((le‘𝑘) × {1o})⟩) = (𝐾 sSet ⟨(Hom ‘ndx), ((le‘𝐾) × {1o})⟩)) |
8 | 7 | oveq1d 7424 | . . . 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 47683 | . . . 4 ⊢ ProsetToCat = (𝑘 ∈ Proset ↦ ((𝑘 sSet ⟨(Hom ‘ndx), ((le‘𝑘) × {1o})⟩) sSet ⟨(comp‘ndx), ∅⟩)) | |
10 | ovex 7442 | . . . 4 ⊢ ((𝐾 sSet ⟨(Hom ‘ndx), ((le‘𝐾) × {1o})⟩) sSet ⟨(comp‘ndx), ∅⟩) ∈ V | |
11 | 8, 9, 10 | fvmpt 6999 | . . 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 2773 | 1 ⊢ (𝜑 → 𝐶 = ((𝐾 sSet ⟨(Hom ‘ndx), ((le‘𝐾) × {1o})⟩) sSet ⟨(comp‘ndx), ∅⟩)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 ∅c0 4323 {csn 4629 ⟨cop 4635 × cxp 5675 ‘cfv 6544 (class class class)co 7409 1oc1o 8459 sSet csts 17096 ndxcnx 17126 lecple 17204 Hom chom 17208 compcco 17209 Proset cproset 18246 ProsetToCatcprstc 47682 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-iota 6496 df-fun 6546 df-fv 6552 df-ov 7412 df-prstc 47683 |
This theorem is referenced by: prstcnidlem 47685 prstcthin 47696 |
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