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Theorem cnvps 18623
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.)
Assertion
Ref Expression
cnvps (𝑅 ∈ PosetRel → 𝑅 ∈ PosetRel)

Proof of Theorem cnvps
StepHypRef Expression
1 relcnv 6122 . . 3 Rel 𝑅
21a1i 11 . 2 (𝑅 ∈ PosetRel → Rel 𝑅)
3 cnvco 5896 . . 3 (𝑅𝑅) = (𝑅𝑅)
4 pstr2 18616 . . . 4 (𝑅 ∈ PosetRel → (𝑅𝑅) ⊆ 𝑅)
5 cnvss 5883 . . . 4 ((𝑅𝑅) ⊆ 𝑅(𝑅𝑅) ⊆ 𝑅)
64, 5syl 17 . . 3 (𝑅 ∈ PosetRel → (𝑅𝑅) ⊆ 𝑅)
73, 6eqsstrrid 4023 . 2 (𝑅 ∈ PosetRel → (𝑅𝑅) ⊆ 𝑅)
8 psrel 18614 . . . . . 6 (𝑅 ∈ PosetRel → Rel 𝑅)
9 dfrel2 6209 . . . . . 6 (Rel 𝑅𝑅 = 𝑅)
108, 9sylib 218 . . . . 5 (𝑅 ∈ PosetRel → 𝑅 = 𝑅)
1110ineq2d 4220 . . . 4 (𝑅 ∈ PosetRel → (𝑅𝑅) = (𝑅𝑅))
12 incom 4209 . . . 4 (𝑅𝑅) = (𝑅𝑅)
1311, 12eqtrdi 2793 . . 3 (𝑅 ∈ PosetRel → (𝑅𝑅) = (𝑅𝑅))
14 psref2 18615 . . 3 (𝑅 ∈ PosetRel → (𝑅𝑅) = ( I ↾ 𝑅))
15 relcnvfld 6300 . . . . 5 (Rel 𝑅 𝑅 = 𝑅)
168, 15syl 17 . . . 4 (𝑅 ∈ PosetRel → 𝑅 = 𝑅)
1716reseq2d 5997 . . 3 (𝑅 ∈ PosetRel → ( I ↾ 𝑅) = ( I ↾ 𝑅))
1813, 14, 173eqtrd 2781 . 2 (𝑅 ∈ PosetRel → (𝑅𝑅) = ( I ↾ 𝑅))
19 cnvexg 7946 . . 3 (𝑅 ∈ PosetRel → 𝑅 ∈ V)
20 isps 18613 . . 3 (𝑅 ∈ V → (𝑅 ∈ PosetRel ↔ (Rel 𝑅 ∧ (𝑅𝑅) ⊆ 𝑅 ∧ (𝑅𝑅) = ( I ↾ 𝑅))))
2119, 20syl 17 . 2 (𝑅 ∈ PosetRel → (𝑅 ∈ PosetRel ↔ (Rel 𝑅 ∧ (𝑅𝑅) ⊆ 𝑅 ∧ (𝑅𝑅) = ( I ↾ 𝑅))))
222, 7, 18, 21mpbir3and 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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