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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 17815 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 5934 | . . 3 ⊢ Rel ◡𝑅 | |
2 | 1 | a1i 11 | . 2 ⊢ (𝑅 ∈ PosetRel → Rel ◡𝑅) |
3 | cnvco 5720 | . . 3 ⊢ ◡(𝑅 ∘ 𝑅) = (◡𝑅 ∘ ◡𝑅) | |
4 | pstr2 17807 | . . . 4 ⊢ (𝑅 ∈ PosetRel → (𝑅 ∘ 𝑅) ⊆ 𝑅) | |
5 | cnvss 5707 | . . . 4 ⊢ ((𝑅 ∘ 𝑅) ⊆ 𝑅 → ◡(𝑅 ∘ 𝑅) ⊆ ◡𝑅) | |
6 | 4, 5 | syl 17 | . . 3 ⊢ (𝑅 ∈ PosetRel → ◡(𝑅 ∘ 𝑅) ⊆ ◡𝑅) |
7 | 3, 6 | eqsstrrid 3964 | . 2 ⊢ (𝑅 ∈ PosetRel → (◡𝑅 ∘ ◡𝑅) ⊆ ◡𝑅) |
8 | psrel 17805 | . . . . . 6 ⊢ (𝑅 ∈ PosetRel → Rel 𝑅) | |
9 | dfrel2 6013 | . . . . . 6 ⊢ (Rel 𝑅 ↔ ◡◡𝑅 = 𝑅) | |
10 | 8, 9 | sylib 221 | . . . . 5 ⊢ (𝑅 ∈ PosetRel → ◡◡𝑅 = 𝑅) |
11 | 10 | ineq2d 4139 | . . . 4 ⊢ (𝑅 ∈ PosetRel → (◡𝑅 ∩ ◡◡𝑅) = (◡𝑅 ∩ 𝑅)) |
12 | incom 4128 | . . . 4 ⊢ (◡𝑅 ∩ 𝑅) = (𝑅 ∩ ◡𝑅) | |
13 | 11, 12 | eqtrdi 2849 | . . 3 ⊢ (𝑅 ∈ PosetRel → (◡𝑅 ∩ ◡◡𝑅) = (𝑅 ∩ ◡𝑅)) |
14 | psref2 17806 | . . 3 ⊢ (𝑅 ∈ PosetRel → (𝑅 ∩ ◡𝑅) = ( I ↾ ∪ ∪ 𝑅)) | |
15 | relcnvfld 6099 | . . . . 5 ⊢ (Rel 𝑅 → ∪ ∪ 𝑅 = ∪ ∪ ◡𝑅) | |
16 | 8, 15 | syl 17 | . . . 4 ⊢ (𝑅 ∈ PosetRel → ∪ ∪ 𝑅 = ∪ ∪ ◡𝑅) |
17 | 16 | reseq2d 5818 | . . 3 ⊢ (𝑅 ∈ PosetRel → ( I ↾ ∪ ∪ 𝑅) = ( I ↾ ∪ ∪ ◡𝑅)) |
18 | 13, 14, 17 | 3eqtrd 2837 | . 2 ⊢ (𝑅 ∈ PosetRel → (◡𝑅 ∩ ◡◡𝑅) = ( I ↾ ∪ ∪ ◡𝑅)) |
19 | cnvexg 7611 | . . 3 ⊢ (𝑅 ∈ PosetRel → ◡𝑅 ∈ V) | |
20 | isps 17804 | . . 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 1339 | 1 ⊢ (𝑅 ∈ PosetRel → ◡𝑅 ∈ PosetRel) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ∩ cin 3880 ⊆ wss 3881 ∪ cuni 4800 I cid 5424 ◡ccnv 5518 ↾ cres 5521 ∘ ccom 5523 Rel wrel 5524 PosetRelcps 17800 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ps 17802 |
This theorem is referenced by: cnvpsb 17815 cnvtsr 17824 ordtcnv 21806 xrge0iifhmeo 31289 |
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