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Theorem cnvps 18116
Description: The converse of a poset is a poset. In the general case (𝑅 ∈ PosetRel → 𝑅 ∈ PosetRel) is not true. See cnvpsb 18117 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 5989 . . 3 Rel 𝑅
21a1i 11 . 2 (𝑅 ∈ PosetRel → Rel 𝑅)
3 cnvco 5771 . . 3 (𝑅𝑅) = (𝑅𝑅)
4 pstr2 18109 . . . 4 (𝑅 ∈ PosetRel → (𝑅𝑅) ⊆ 𝑅)
5 cnvss 5758 . . . 4 ((𝑅𝑅) ⊆ 𝑅(𝑅𝑅) ⊆ 𝑅)
64, 5syl 17 . . 3 (𝑅 ∈ PosetRel → (𝑅𝑅) ⊆ 𝑅)
73, 6eqsstrrid 3966 . 2 (𝑅 ∈ PosetRel → (𝑅𝑅) ⊆ 𝑅)
8 psrel 18107 . . . . . 6 (𝑅 ∈ PosetRel → Rel 𝑅)
9 dfrel2 6069 . . . . . 6 (Rel 𝑅𝑅 = 𝑅)
108, 9sylib 221 . . . . 5 (𝑅 ∈ PosetRel → 𝑅 = 𝑅)
1110ineq2d 4143 . . . 4 (𝑅 ∈ PosetRel → (𝑅𝑅) = (𝑅𝑅))
12 incom 4131 . . . 4 (𝑅𝑅) = (𝑅𝑅)
1311, 12eqtrdi 2796 . . 3 (𝑅 ∈ PosetRel → (𝑅𝑅) = (𝑅𝑅))
14 psref2 18108 . . 3 (𝑅 ∈ PosetRel → (𝑅𝑅) = ( I ↾ 𝑅))
15 relcnvfld 6160 . . . . 5 (Rel 𝑅 𝑅 = 𝑅)
168, 15syl 17 . . . 4 (𝑅 ∈ PosetRel → 𝑅 = 𝑅)
1716reseq2d 5868 . . 3 (𝑅 ∈ PosetRel → ( I ↾ 𝑅) = ( I ↾ 𝑅))
1813, 14, 173eqtrd 2783 . 2 (𝑅 ∈ PosetRel → (𝑅𝑅) = ( I ↾ 𝑅))
19 cnvexg 7723 . . 3 (𝑅 ∈ PosetRel → 𝑅 ∈ V)
20 isps 18106 . . 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 1344 1 (𝑅 ∈ PosetRel → 𝑅 ∈ PosetRel)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  w3a 1089   = wceq 1543  wcel 2112  Vcvv 3423  cin 3882  wss 3883   cuni 4835   I cid 5470  ccnv 5567  cres 5570  ccom 5572  Rel wrel 5573  PosetRelcps 18102
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2114  ax-9 2122  ax-12 2177  ax-ext 2710  ax-sep 5208  ax-nul 5215  ax-pow 5274  ax-pr 5338  ax-un 7544
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2073  df-clab 2717  df-cleq 2731  df-clel 2818  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3425  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4456  df-pw 4531  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4836  df-br 5070  df-opab 5132  df-xp 5574  df-rel 5575  df-cnv 5576  df-co 5577  df-dm 5578  df-rn 5579  df-res 5580  df-ps 18104
This theorem is referenced by:  cnvpsb  18117  cnvtsr  18126  ordtcnv  22129  xrge0iifhmeo  31631
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