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Mirrors > Home > MPE Home > Th. List > isps | Structured version Visualization version GIF version |
Description: The predicate "is a poset" i.e. a transitive, reflexive, antisymmetric relation. (Contributed by NM, 11-May-2008.) |
Ref | Expression |
---|---|
isps | ⊢ (𝑅 ∈ 𝐴 → (𝑅 ∈ PosetRel ↔ (Rel 𝑅 ∧ (𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ (𝑅 ∩ ◡𝑅) = ( I ↾ ∪ ∪ 𝑅)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | releq 5644 | . . 3 ⊢ (𝑟 = 𝑅 → (Rel 𝑟 ↔ Rel 𝑅)) | |
2 | coeq1 5721 | . . . . 5 ⊢ (𝑟 = 𝑅 → (𝑟 ∘ 𝑟) = (𝑅 ∘ 𝑟)) | |
3 | coeq2 5722 | . . . . 5 ⊢ (𝑟 = 𝑅 → (𝑅 ∘ 𝑟) = (𝑅 ∘ 𝑅)) | |
4 | 2, 3 | eqtrd 2853 | . . . 4 ⊢ (𝑟 = 𝑅 → (𝑟 ∘ 𝑟) = (𝑅 ∘ 𝑅)) |
5 | id 22 | . . . 4 ⊢ (𝑟 = 𝑅 → 𝑟 = 𝑅) | |
6 | 4, 5 | sseq12d 3997 | . . 3 ⊢ (𝑟 = 𝑅 → ((𝑟 ∘ 𝑟) ⊆ 𝑟 ↔ (𝑅 ∘ 𝑅) ⊆ 𝑅)) |
7 | cnveq 5737 | . . . . 5 ⊢ (𝑟 = 𝑅 → ◡𝑟 = ◡𝑅) | |
8 | 5, 7 | ineq12d 4187 | . . . 4 ⊢ (𝑟 = 𝑅 → (𝑟 ∩ ◡𝑟) = (𝑅 ∩ ◡𝑅)) |
9 | unieq 4838 | . . . . . 6 ⊢ (𝑟 = 𝑅 → ∪ 𝑟 = ∪ 𝑅) | |
10 | 9 | unieqd 4840 | . . . . 5 ⊢ (𝑟 = 𝑅 → ∪ ∪ 𝑟 = ∪ ∪ 𝑅) |
11 | 10 | reseq2d 5846 | . . . 4 ⊢ (𝑟 = 𝑅 → ( I ↾ ∪ ∪ 𝑟) = ( I ↾ ∪ ∪ 𝑅)) |
12 | 8, 11 | eqeq12d 2834 | . . 3 ⊢ (𝑟 = 𝑅 → ((𝑟 ∩ ◡𝑟) = ( I ↾ ∪ ∪ 𝑟) ↔ (𝑅 ∩ ◡𝑅) = ( I ↾ ∪ ∪ 𝑅))) |
13 | 1, 6, 12 | 3anbi123d 1427 | . 2 ⊢ (𝑟 = 𝑅 → ((Rel 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟 ∧ (𝑟 ∩ ◡𝑟) = ( I ↾ ∪ ∪ 𝑟)) ↔ (Rel 𝑅 ∧ (𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ (𝑅 ∩ ◡𝑅) = ( I ↾ ∪ ∪ 𝑅)))) |
14 | df-ps 17798 | . 2 ⊢ PosetRel = {𝑟 ∣ (Rel 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟 ∧ (𝑟 ∩ ◡𝑟) = ( I ↾ ∪ ∪ 𝑟))} | |
15 | 13, 14 | elab2g 3665 | 1 ⊢ (𝑅 ∈ 𝐴 → (𝑅 ∈ PosetRel ↔ (Rel 𝑅 ∧ (𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ (𝑅 ∩ ◡𝑅) = ( I ↾ ∪ ∪ 𝑅)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ∩ cin 3932 ⊆ wss 3933 ∪ cuni 4830 I cid 5452 ◡ccnv 5547 ↾ cres 5550 ∘ ccom 5552 Rel wrel 5553 PosetRelcps 17796 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-rex 3141 df-rab 3144 df-in 3940 df-ss 3949 df-uni 4831 df-br 5058 df-opab 5120 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-res 5560 df-ps 17798 |
This theorem is referenced by: psrel 17801 psref2 17802 pstr2 17803 cnvps 17810 psss 17812 letsr 17825 |
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