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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 5772 | . . 3 ⊢ (𝑟 = 𝑅 → (Rel 𝑟 ↔ Rel 𝑅)) | |
2 | coeq1 5854 | . . . . 5 ⊢ (𝑟 = 𝑅 → (𝑟 ∘ 𝑟) = (𝑅 ∘ 𝑟)) | |
3 | coeq2 5855 | . . . . 5 ⊢ (𝑟 = 𝑅 → (𝑅 ∘ 𝑟) = (𝑅 ∘ 𝑅)) | |
4 | 2, 3 | eqtrd 2768 | . . . 4 ⊢ (𝑟 = 𝑅 → (𝑟 ∘ 𝑟) = (𝑅 ∘ 𝑅)) |
5 | id 22 | . . . 4 ⊢ (𝑟 = 𝑅 → 𝑟 = 𝑅) | |
6 | 4, 5 | sseq12d 4011 | . . 3 ⊢ (𝑟 = 𝑅 → ((𝑟 ∘ 𝑟) ⊆ 𝑟 ↔ (𝑅 ∘ 𝑅) ⊆ 𝑅)) |
7 | cnveq 5870 | . . . . 5 ⊢ (𝑟 = 𝑅 → ◡𝑟 = ◡𝑅) | |
8 | 5, 7 | ineq12d 4209 | . . . 4 ⊢ (𝑟 = 𝑅 → (𝑟 ∩ ◡𝑟) = (𝑅 ∩ ◡𝑅)) |
9 | unieq 4914 | . . . . . 6 ⊢ (𝑟 = 𝑅 → ∪ 𝑟 = ∪ 𝑅) | |
10 | 9 | unieqd 4916 | . . . . 5 ⊢ (𝑟 = 𝑅 → ∪ ∪ 𝑟 = ∪ ∪ 𝑅) |
11 | 10 | reseq2d 5979 | . . . 4 ⊢ (𝑟 = 𝑅 → ( I ↾ ∪ ∪ 𝑟) = ( I ↾ ∪ ∪ 𝑅)) |
12 | 8, 11 | eqeq12d 2744 | . . 3 ⊢ (𝑟 = 𝑅 → ((𝑟 ∩ ◡𝑟) = ( I ↾ ∪ ∪ 𝑟) ↔ (𝑅 ∩ ◡𝑅) = ( I ↾ ∪ ∪ 𝑅))) |
13 | 1, 6, 12 | 3anbi123d 1433 | . 2 ⊢ (𝑟 = 𝑅 → ((Rel 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟 ∧ (𝑟 ∩ ◡𝑟) = ( I ↾ ∪ ∪ 𝑟)) ↔ (Rel 𝑅 ∧ (𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ (𝑅 ∩ ◡𝑅) = ( I ↾ ∪ ∪ 𝑅)))) |
14 | df-ps 18551 | . 2 ⊢ PosetRel = {𝑟 ∣ (Rel 𝑟 ∧ (𝑟 ∘ 𝑟) ⊆ 𝑟 ∧ (𝑟 ∩ ◡𝑟) = ( I ↾ ∪ ∪ 𝑟))} | |
15 | 13, 14 | elab2g 3668 | 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 205 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ∩ cin 3944 ⊆ wss 3945 ∪ cuni 4903 I cid 5569 ◡ccnv 5671 ↾ cres 5674 ∘ ccom 5676 Rel wrel 5677 PosetRelcps 18549 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 396 df-3an 1087 df-tru 1537 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-rab 3429 df-v 3472 df-in 3952 df-ss 3962 df-uni 4904 df-br 5143 df-opab 5205 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-res 5684 df-ps 18551 |
This theorem is referenced by: psrel 18554 psref2 18555 pstr2 18556 cnvps 18563 psss 18565 letsr 18578 |
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