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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 5648 | . . 3 ⊢ (𝑟 = 𝑅 → (Rel 𝑟 ↔ Rel 𝑅)) | |
2 | coeq1 5726 | . . . . 5 ⊢ (𝑟 = 𝑅 → (𝑟 ∘ 𝑟) = (𝑅 ∘ 𝑟)) | |
3 | coeq2 5727 | . . . . 5 ⊢ (𝑟 = 𝑅 → (𝑅 ∘ 𝑟) = (𝑅 ∘ 𝑅)) | |
4 | 2, 3 | eqtrd 2777 | . . . 4 ⊢ (𝑟 = 𝑅 → (𝑟 ∘ 𝑟) = (𝑅 ∘ 𝑅)) |
5 | id 22 | . . . 4 ⊢ (𝑟 = 𝑅 → 𝑟 = 𝑅) | |
6 | 4, 5 | sseq12d 3934 | . . 3 ⊢ (𝑟 = 𝑅 → ((𝑟 ∘ 𝑟) ⊆ 𝑟 ↔ (𝑅 ∘ 𝑅) ⊆ 𝑅)) |
7 | cnveq 5742 | . . . . 5 ⊢ (𝑟 = 𝑅 → ◡𝑟 = ◡𝑅) | |
8 | 5, 7 | ineq12d 4128 | . . . 4 ⊢ (𝑟 = 𝑅 → (𝑟 ∩ ◡𝑟) = (𝑅 ∩ ◡𝑅)) |
9 | unieq 4830 | . . . . . 6 ⊢ (𝑟 = 𝑅 → ∪ 𝑟 = ∪ 𝑅) | |
10 | 9 | unieqd 4833 | . . . . 5 ⊢ (𝑟 = 𝑅 → ∪ ∪ 𝑟 = ∪ ∪ 𝑅) |
11 | 10 | reseq2d 5851 | . . . 4 ⊢ (𝑟 = 𝑅 → ( I ↾ ∪ ∪ 𝑟) = ( I ↾ ∪ ∪ 𝑅)) |
12 | 8, 11 | eqeq12d 2753 | . . 3 ⊢ (𝑟 = 𝑅 → ((𝑟 ∩ ◡𝑟) = ( I ↾ ∪ ∪ 𝑟) ↔ (𝑅 ∩ ◡𝑅) = ( I ↾ ∪ ∪ 𝑅))) |
13 | 1, 6, 12 | 3anbi123d 1438 | . 2 ⊢ (𝑟 = 𝑅 → ((Rel 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟 ∧ (𝑟 ∩ ◡𝑟) = ( I ↾ ∪ ∪ 𝑟)) ↔ (Rel 𝑅 ∧ (𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ (𝑅 ∩ ◡𝑅) = ( I ↾ ∪ ∪ 𝑅)))) |
14 | df-ps 18072 | . 2 ⊢ PosetRel = {𝑟 ∣ (Rel 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟 ∧ (𝑟 ∩ ◡𝑟) = ( I ↾ ∪ ∪ 𝑟))} | |
15 | 13, 14 | elab2g 3589 | 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 209 ∧ w3a 1089 = wceq 1543 ∈ wcel 2110 ∩ cin 3865 ⊆ wss 3866 ∪ cuni 4819 I cid 5454 ◡ccnv 5550 ↾ cres 5553 ∘ ccom 5555 Rel wrel 5556 PosetRelcps 18070 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-3an 1091 df-tru 1546 df-ex 1788 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3070 df-v 3410 df-in 3873 df-ss 3883 df-uni 4820 df-br 5054 df-opab 5116 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-res 5563 df-ps 18072 |
This theorem is referenced by: psrel 18075 psref2 18076 pstr2 18077 cnvps 18084 psss 18086 letsr 18099 |
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