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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 18624 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 6122 | . . 3 ⊢ Rel ◡𝑅 | |
| 2 | 1 | a1i 11 | . 2 ⊢ (𝑅 ∈ PosetRel → Rel ◡𝑅) |
| 3 | cnvco 5896 | . . 3 ⊢ ◡(𝑅 ∘ 𝑅) = (◡𝑅 ∘ ◡𝑅) | |
| 4 | pstr2 18616 | . . . 4 ⊢ (𝑅 ∈ PosetRel → (𝑅 ∘ 𝑅) ⊆ 𝑅) | |
| 5 | cnvss 5883 | . . . 4 ⊢ ((𝑅 ∘ 𝑅) ⊆ 𝑅 → ◡(𝑅 ∘ 𝑅) ⊆ ◡𝑅) | |
| 6 | 4, 5 | syl 17 | . . 3 ⊢ (𝑅 ∈ PosetRel → ◡(𝑅 ∘ 𝑅) ⊆ ◡𝑅) |
| 7 | 3, 6 | eqsstrrid 4023 | . 2 ⊢ (𝑅 ∈ PosetRel → (◡𝑅 ∘ ◡𝑅) ⊆ ◡𝑅) |
| 8 | psrel 18614 | . . . . . 6 ⊢ (𝑅 ∈ PosetRel → Rel 𝑅) | |
| 9 | dfrel2 6209 | . . . . . 6 ⊢ (Rel 𝑅 ↔ ◡◡𝑅 = 𝑅) | |
| 10 | 8, 9 | sylib 218 | . . . . 5 ⊢ (𝑅 ∈ PosetRel → ◡◡𝑅 = 𝑅) |
| 11 | 10 | ineq2d 4220 | . . . 4 ⊢ (𝑅 ∈ PosetRel → (◡𝑅 ∩ ◡◡𝑅) = (◡𝑅 ∩ 𝑅)) |
| 12 | incom 4209 | . . . 4 ⊢ (◡𝑅 ∩ 𝑅) = (𝑅 ∩ ◡𝑅) | |
| 13 | 11, 12 | eqtrdi 2793 | . . 3 ⊢ (𝑅 ∈ PosetRel → (◡𝑅 ∩ ◡◡𝑅) = (𝑅 ∩ ◡𝑅)) |
| 14 | psref2 18615 | . . 3 ⊢ (𝑅 ∈ PosetRel → (𝑅 ∩ ◡𝑅) = ( I ↾ ∪ ∪ 𝑅)) | |
| 15 | relcnvfld 6300 | . . . . 5 ⊢ (Rel 𝑅 → ∪ ∪ 𝑅 = ∪ ∪ ◡𝑅) | |
| 16 | 8, 15 | syl 17 | . . . 4 ⊢ (𝑅 ∈ PosetRel → ∪ ∪ 𝑅 = ∪ ∪ ◡𝑅) |
| 17 | 16 | reseq2d 5997 | . . 3 ⊢ (𝑅 ∈ PosetRel → ( I ↾ ∪ ∪ 𝑅) = ( I ↾ ∪ ∪ ◡𝑅)) |
| 18 | 13, 14, 17 | 3eqtrd 2781 | . 2 ⊢ (𝑅 ∈ PosetRel → (◡𝑅 ∩ ◡◡𝑅) = ( I ↾ ∪ ∪ ◡𝑅)) |
| 19 | cnvexg 7946 | . . 3 ⊢ (𝑅 ∈ PosetRel → ◡𝑅 ∈ V) | |
| 20 | isps 18613 | . . 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 1343 | 1 ⊢ (𝑅 ∈ PosetRel → ◡𝑅 ∈ PosetRel) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∩ cin 3950 ⊆ wss 3951 ∪ cuni 4907 I cid 5577 ◡ccnv 5684 ↾ cres 5687 ∘ ccom 5689 Rel wrel 5690 PosetRelcps 18609 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ps 18611 |
| This theorem is referenced by: cnvpsb 18624 cnvtsr 18633 ordtcnv 23209 xrge0iifhmeo 33935 |
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