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Theorem cnvps 17334
Description: The converse of a poset is a poset. In the general case (𝑅 ∈ PosetRel → 𝑅 ∈ PosetRel) is not true. See cnvpsb 17335 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 5613 . . 3 Rel 𝑅
21a1i 11 . 2 (𝑅 ∈ PosetRel → Rel 𝑅)
3 cnvco 5415 . . 3 (𝑅𝑅) = (𝑅𝑅)
4 pstr2 17327 . . . 4 (𝑅 ∈ PosetRel → (𝑅𝑅) ⊆ 𝑅)
5 cnvss 5402 . . . 4 ((𝑅𝑅) ⊆ 𝑅(𝑅𝑅) ⊆ 𝑅)
64, 5syl 17 . . 3 (𝑅 ∈ PosetRel → (𝑅𝑅) ⊆ 𝑅)
73, 6syl5eqssr 3756 . 2 (𝑅 ∈ PosetRel → (𝑅𝑅) ⊆ 𝑅)
8 psrel 17325 . . . . . 6 (𝑅 ∈ PosetRel → Rel 𝑅)
9 dfrel2 5693 . . . . . 6 (Rel 𝑅𝑅 = 𝑅)
108, 9sylib 208 . . . . 5 (𝑅 ∈ PosetRel → 𝑅 = 𝑅)
1110ineq2d 3922 . . . 4 (𝑅 ∈ PosetRel → (𝑅𝑅) = (𝑅𝑅))
12 incom 3913 . . . 4 (𝑅𝑅) = (𝑅𝑅)
1311, 12syl6eq 2774 . . 3 (𝑅 ∈ PosetRel → (𝑅𝑅) = (𝑅𝑅))
14 psref2 17326 . . 3 (𝑅 ∈ PosetRel → (𝑅𝑅) = ( I ↾ 𝑅))
15 relcnvfld 5779 . . . . 5 (Rel 𝑅 𝑅 = 𝑅)
168, 15syl 17 . . . 4 (𝑅 ∈ PosetRel → 𝑅 = 𝑅)
1716reseq2d 5503 . . 3 (𝑅 ∈ PosetRel → ( I ↾ 𝑅) = ( I ↾ 𝑅))
1813, 14, 173eqtrd 2762 . 2 (𝑅 ∈ PosetRel → (𝑅𝑅) = ( I ↾ 𝑅))
19 cnvexg 7229 . . 3 (𝑅 ∈ PosetRel → 𝑅 ∈ V)
20 isps 17324 . . 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 1382 1 (𝑅 ∈ PosetRel → 𝑅 ∈ PosetRel)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  w3a 1072   = wceq 1596  wcel 2103  Vcvv 3304  cin 3679  wss 3680   cuni 4544   I cid 5127  ccnv 5217  cres 5220  ccom 5222  Rel wrel 5223  PosetRelcps 17320
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ral 3019  df-rex 3020  df-rab 3023  df-v 3306  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-br 4761  df-opab 4821  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ps 17322
This theorem is referenced by:  cnvpsb  17335  cnvtsr  17344  ordtcnv  21128  xrge0iifhmeo  30212
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