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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 17036 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 5422 | . . 3 ⊢ Rel ◡𝑅 | |
2 | 1 | a1i 11 | . 2 ⊢ (𝑅 ∈ PosetRel → Rel ◡𝑅) |
3 | cnvco 5230 | . . 3 ⊢ ◡(𝑅 ∘ 𝑅) = (◡𝑅 ∘ ◡𝑅) | |
4 | pstr2 17028 | . . . 4 ⊢ (𝑅 ∈ PosetRel → (𝑅 ∘ 𝑅) ⊆ 𝑅) | |
5 | cnvss 5216 | . . . 4 ⊢ ((𝑅 ∘ 𝑅) ⊆ 𝑅 → ◡(𝑅 ∘ 𝑅) ⊆ ◡𝑅) | |
6 | 4, 5 | syl 17 | . . 3 ⊢ (𝑅 ∈ PosetRel → ◡(𝑅 ∘ 𝑅) ⊆ ◡𝑅) |
7 | 3, 6 | syl5eqssr 3613 | . 2 ⊢ (𝑅 ∈ PosetRel → (◡𝑅 ∘ ◡𝑅) ⊆ ◡𝑅) |
8 | psrel 17026 | . . . . . 6 ⊢ (𝑅 ∈ PosetRel → Rel 𝑅) | |
9 | dfrel2 5502 | . . . . . 6 ⊢ (Rel 𝑅 ↔ ◡◡𝑅 = 𝑅) | |
10 | 8, 9 | sylib 207 | . . . . 5 ⊢ (𝑅 ∈ PosetRel → ◡◡𝑅 = 𝑅) |
11 | 10 | ineq2d 3776 | . . . 4 ⊢ (𝑅 ∈ PosetRel → (◡𝑅 ∩ ◡◡𝑅) = (◡𝑅 ∩ 𝑅)) |
12 | incom 3767 | . . . 4 ⊢ (◡𝑅 ∩ 𝑅) = (𝑅 ∩ ◡𝑅) | |
13 | 11, 12 | syl6eq 2660 | . . 3 ⊢ (𝑅 ∈ PosetRel → (◡𝑅 ∩ ◡◡𝑅) = (𝑅 ∩ ◡𝑅)) |
14 | psref2 17027 | . . 3 ⊢ (𝑅 ∈ PosetRel → (𝑅 ∩ ◡𝑅) = ( I ↾ ∪ ∪ 𝑅)) | |
15 | relcnvfld 5583 | . . . . 5 ⊢ (Rel 𝑅 → ∪ ∪ 𝑅 = ∪ ∪ ◡𝑅) | |
16 | 8, 15 | syl 17 | . . . 4 ⊢ (𝑅 ∈ PosetRel → ∪ ∪ 𝑅 = ∪ ∪ ◡𝑅) |
17 | 16 | reseq2d 5317 | . . 3 ⊢ (𝑅 ∈ PosetRel → ( I ↾ ∪ ∪ 𝑅) = ( I ↾ ∪ ∪ ◡𝑅)) |
18 | 13, 14, 17 | 3eqtrd 2648 | . 2 ⊢ (𝑅 ∈ PosetRel → (◡𝑅 ∩ ◡◡𝑅) = ( I ↾ ∪ ∪ ◡𝑅)) |
19 | cnvexg 7005 | . . 3 ⊢ (𝑅 ∈ PosetRel → ◡𝑅 ∈ V) | |
20 | isps 17025 | . . 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 1238 | 1 ⊢ (𝑅 ∈ PosetRel → ◡𝑅 ∈ PosetRel) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∩ cin 3539 ⊆ wss 3540 ∪ cuni 4372 I cid 4948 ◡ccnv 5037 ↾ cres 5040 ∘ ccom 5042 Rel wrel 5043 PosetRelcps 17021 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ps 17023 |
This theorem is referenced by: cnvpsb 17036 cnvtsr 17045 ordtcnv 20815 xrge0iifhmeo 29310 |
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