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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 18117 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 5989 | . . 3 ⊢ Rel ◡𝑅 | |
2 | 1 | a1i 11 | . 2 ⊢ (𝑅 ∈ PosetRel → Rel ◡𝑅) |
3 | cnvco 5771 | . . 3 ⊢ ◡(𝑅 ∘ 𝑅) = (◡𝑅 ∘ ◡𝑅) | |
4 | pstr2 18109 | . . . 4 ⊢ (𝑅 ∈ PosetRel → (𝑅 ∘ 𝑅) ⊆ 𝑅) | |
5 | cnvss 5758 | . . . 4 ⊢ ((𝑅 ∘ 𝑅) ⊆ 𝑅 → ◡(𝑅 ∘ 𝑅) ⊆ ◡𝑅) | |
6 | 4, 5 | syl 17 | . . 3 ⊢ (𝑅 ∈ PosetRel → ◡(𝑅 ∘ 𝑅) ⊆ ◡𝑅) |
7 | 3, 6 | eqsstrrid 3966 | . 2 ⊢ (𝑅 ∈ PosetRel → (◡𝑅 ∘ ◡𝑅) ⊆ ◡𝑅) |
8 | psrel 18107 | . . . . . 6 ⊢ (𝑅 ∈ PosetRel → Rel 𝑅) | |
9 | dfrel2 6069 | . . . . . 6 ⊢ (Rel 𝑅 ↔ ◡◡𝑅 = 𝑅) | |
10 | 8, 9 | sylib 221 | . . . . 5 ⊢ (𝑅 ∈ PosetRel → ◡◡𝑅 = 𝑅) |
11 | 10 | ineq2d 4143 | . . . 4 ⊢ (𝑅 ∈ PosetRel → (◡𝑅 ∩ ◡◡𝑅) = (◡𝑅 ∩ 𝑅)) |
12 | incom 4131 | . . . 4 ⊢ (◡𝑅 ∩ 𝑅) = (𝑅 ∩ ◡𝑅) | |
13 | 11, 12 | eqtrdi 2796 | . . 3 ⊢ (𝑅 ∈ PosetRel → (◡𝑅 ∩ ◡◡𝑅) = (𝑅 ∩ ◡𝑅)) |
14 | psref2 18108 | . . 3 ⊢ (𝑅 ∈ PosetRel → (𝑅 ∩ ◡𝑅) = ( I ↾ ∪ ∪ 𝑅)) | |
15 | relcnvfld 6160 | . . . . 5 ⊢ (Rel 𝑅 → ∪ ∪ 𝑅 = ∪ ∪ ◡𝑅) | |
16 | 8, 15 | syl 17 | . . . 4 ⊢ (𝑅 ∈ PosetRel → ∪ ∪ 𝑅 = ∪ ∪ ◡𝑅) |
17 | 16 | reseq2d 5868 | . . 3 ⊢ (𝑅 ∈ PosetRel → ( I ↾ ∪ ∪ 𝑅) = ( I ↾ ∪ ∪ ◡𝑅)) |
18 | 13, 14, 17 | 3eqtrd 2783 | . 2 ⊢ (𝑅 ∈ PosetRel → (◡𝑅 ∩ ◡◡𝑅) = ( I ↾ ∪ ∪ ◡𝑅)) |
19 | cnvexg 7723 | . . 3 ⊢ (𝑅 ∈ PosetRel → ◡𝑅 ∈ V) | |
20 | isps 18106 | . . 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 1344 | 1 ⊢ (𝑅 ∈ PosetRel → ◡𝑅 ∈ PosetRel) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ w3a 1089 = wceq 1543 ∈ wcel 2112 Vcvv 3423 ∩ cin 3882 ⊆ wss 3883 ∪ cuni 4835 I cid 5470 ◡ccnv 5567 ↾ cres 5570 ∘ ccom 5572 Rel wrel 5573 PosetRelcps 18102 |
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 2114 ax-9 2122 ax-12 2177 ax-ext 2710 ax-sep 5208 ax-nul 5215 ax-pow 5274 ax-pr 5338 ax-un 7544 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2073 df-clab 2717 df-cleq 2731 df-clel 2818 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3425 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4254 df-if 4456 df-pw 4531 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4836 df-br 5070 df-opab 5132 df-xp 5574 df-rel 5575 df-cnv 5576 df-co 5577 df-dm 5578 df-rn 5579 df-res 5580 df-ps 18104 |
This theorem is referenced by: cnvpsb 18117 cnvtsr 18126 ordtcnv 22129 xrge0iifhmeo 31631 |
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