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Mirrors > Home > MPE Home > Th. List > cnvps | Structured version Visualization version GIF version |
Description: The converse of a poset is a poset. In the general case (◡𝑅 ∈ PosetRel → 𝑅 ∈ PosetRel) is not true. See cnvpsb 18637 for a special case where the property holds. (Contributed by FL, 5-Jan-2009.) (Proof shortened by Mario Carneiro, 3-Sep-2015.) |
Ref | Expression |
---|---|
cnvps | ⊢ (𝑅 ∈ PosetRel → ◡𝑅 ∈ PosetRel) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relcnv 6125 | . . 3 ⊢ Rel ◡𝑅 | |
2 | 1 | a1i 11 | . 2 ⊢ (𝑅 ∈ PosetRel → Rel ◡𝑅) |
3 | cnvco 5899 | . . 3 ⊢ ◡(𝑅 ∘ 𝑅) = (◡𝑅 ∘ ◡𝑅) | |
4 | pstr2 18629 | . . . 4 ⊢ (𝑅 ∈ PosetRel → (𝑅 ∘ 𝑅) ⊆ 𝑅) | |
5 | cnvss 5886 | . . . 4 ⊢ ((𝑅 ∘ 𝑅) ⊆ 𝑅 → ◡(𝑅 ∘ 𝑅) ⊆ ◡𝑅) | |
6 | 4, 5 | syl 17 | . . 3 ⊢ (𝑅 ∈ PosetRel → ◡(𝑅 ∘ 𝑅) ⊆ ◡𝑅) |
7 | 3, 6 | eqsstrrid 4045 | . 2 ⊢ (𝑅 ∈ PosetRel → (◡𝑅 ∘ ◡𝑅) ⊆ ◡𝑅) |
8 | psrel 18627 | . . . . . 6 ⊢ (𝑅 ∈ PosetRel → Rel 𝑅) | |
9 | dfrel2 6211 | . . . . . 6 ⊢ (Rel 𝑅 ↔ ◡◡𝑅 = 𝑅) | |
10 | 8, 9 | sylib 218 | . . . . 5 ⊢ (𝑅 ∈ PosetRel → ◡◡𝑅 = 𝑅) |
11 | 10 | ineq2d 4228 | . . . 4 ⊢ (𝑅 ∈ PosetRel → (◡𝑅 ∩ ◡◡𝑅) = (◡𝑅 ∩ 𝑅)) |
12 | incom 4217 | . . . 4 ⊢ (◡𝑅 ∩ 𝑅) = (𝑅 ∩ ◡𝑅) | |
13 | 11, 12 | eqtrdi 2791 | . . 3 ⊢ (𝑅 ∈ PosetRel → (◡𝑅 ∩ ◡◡𝑅) = (𝑅 ∩ ◡𝑅)) |
14 | psref2 18628 | . . 3 ⊢ (𝑅 ∈ PosetRel → (𝑅 ∩ ◡𝑅) = ( I ↾ ∪ ∪ 𝑅)) | |
15 | relcnvfld 6302 | . . . . 5 ⊢ (Rel 𝑅 → ∪ ∪ 𝑅 = ∪ ∪ ◡𝑅) | |
16 | 8, 15 | syl 17 | . . . 4 ⊢ (𝑅 ∈ PosetRel → ∪ ∪ 𝑅 = ∪ ∪ ◡𝑅) |
17 | 16 | reseq2d 6000 | . . 3 ⊢ (𝑅 ∈ PosetRel → ( I ↾ ∪ ∪ 𝑅) = ( I ↾ ∪ ∪ ◡𝑅)) |
18 | 13, 14, 17 | 3eqtrd 2779 | . 2 ⊢ (𝑅 ∈ PosetRel → (◡𝑅 ∩ ◡◡𝑅) = ( I ↾ ∪ ∪ ◡𝑅)) |
19 | cnvexg 7947 | . . 3 ⊢ (𝑅 ∈ PosetRel → ◡𝑅 ∈ V) | |
20 | isps 18626 | . . 3 ⊢ (◡𝑅 ∈ V → (◡𝑅 ∈ PosetRel ↔ (Rel ◡𝑅 ∧ (◡𝑅 ∘ ◡𝑅) ⊆ ◡𝑅 ∧ (◡𝑅 ∩ ◡◡𝑅) = ( I ↾ ∪ ∪ ◡𝑅)))) | |
21 | 19, 20 | syl 17 | . 2 ⊢ (𝑅 ∈ PosetRel → (◡𝑅 ∈ PosetRel ↔ (Rel ◡𝑅 ∧ (◡𝑅 ∘ ◡𝑅) ⊆ ◡𝑅 ∧ (◡𝑅 ∩ ◡◡𝑅) = ( I ↾ ∪ ∪ ◡𝑅)))) |
22 | 2, 7, 18, 21 | mpbir3and 1341 | 1 ⊢ (𝑅 ∈ PosetRel → ◡𝑅 ∈ PosetRel) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 Vcvv 3478 ∩ cin 3962 ⊆ wss 3963 ∪ cuni 4912 I cid 5582 ◡ccnv 5688 ↾ cres 5691 ∘ ccom 5693 Rel wrel 5694 PosetRelcps 18622 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ps 18624 |
This theorem is referenced by: cnvpsb 18637 cnvtsr 18646 ordtcnv 23225 xrge0iifhmeo 33897 |
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