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Theorem cnvps 17035
 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.)
Assertion
Ref Expression
cnvps (𝑅 ∈ PosetRel → 𝑅 ∈ PosetRel)

Proof of Theorem cnvps
StepHypRef Expression
1 relcnv 5422 . . 3 Rel 𝑅
21a1i 11 . 2 (𝑅 ∈ PosetRel → Rel 𝑅)
3 cnvco 5230 . . 3 (𝑅𝑅) = (𝑅𝑅)
4 pstr2 17028 . . . 4 (𝑅 ∈ PosetRel → (𝑅𝑅) ⊆ 𝑅)
5 cnvss 5216 . . . 4 ((𝑅𝑅) ⊆ 𝑅(𝑅𝑅) ⊆ 𝑅)
64, 5syl 17 . . 3 (𝑅 ∈ PosetRel → (𝑅𝑅) ⊆ 𝑅)
73, 6syl5eqssr 3613 . 2 (𝑅 ∈ PosetRel → (𝑅𝑅) ⊆ 𝑅)
8 psrel 17026 . . . . . 6 (𝑅 ∈ PosetRel → Rel 𝑅)
9 dfrel2 5502 . . . . . 6 (Rel 𝑅𝑅 = 𝑅)
108, 9sylib 207 . . . . 5 (𝑅 ∈ PosetRel → 𝑅 = 𝑅)
1110ineq2d 3776 . . . 4 (𝑅 ∈ PosetRel → (𝑅𝑅) = (𝑅𝑅))
12 incom 3767 . . . 4 (𝑅𝑅) = (𝑅𝑅)
1311, 12syl6eq 2660 . . 3 (𝑅 ∈ PosetRel → (𝑅𝑅) = (𝑅𝑅))
14 psref2 17027 . . 3 (𝑅 ∈ PosetRel → (𝑅𝑅) = ( I ↾ 𝑅))
15 relcnvfld 5583 . . . . 5 (Rel 𝑅 𝑅 = 𝑅)
168, 15syl 17 . . . 4 (𝑅 ∈ PosetRel → 𝑅 = 𝑅)
1716reseq2d 5317 . . 3 (𝑅 ∈ PosetRel → ( I ↾ 𝑅) = ( I ↾ 𝑅))
1813, 14, 173eqtrd 2648 . 2 (𝑅 ∈ PosetRel → (𝑅𝑅) = ( I ↾ 𝑅))
19 cnvexg 7005 . . 3 (𝑅 ∈ PosetRel → 𝑅 ∈ V)
20 isps 17025 . . 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 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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