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Theorem cnvps 17814
 Description: The converse of a poset is a poset. In the general case (◡𝑅 ∈ PosetRel → 𝑅 ∈ PosetRel) is not true. See cnvpsb 17815 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 5960 . . 3 Rel 𝑅
21a1i 11 . 2 (𝑅 ∈ PosetRel → Rel 𝑅)
3 cnvco 5749 . . 3 (𝑅𝑅) = (𝑅𝑅)
4 pstr2 17807 . . . 4 (𝑅 ∈ PosetRel → (𝑅𝑅) ⊆ 𝑅)
5 cnvss 5736 . . . 4 ((𝑅𝑅) ⊆ 𝑅(𝑅𝑅) ⊆ 𝑅)
64, 5syl 17 . . 3 (𝑅 ∈ PosetRel → (𝑅𝑅) ⊆ 𝑅)
73, 6eqsstrrid 4014 . 2 (𝑅 ∈ PosetRel → (𝑅𝑅) ⊆ 𝑅)
8 psrel 17805 . . . . . 6 (𝑅 ∈ PosetRel → Rel 𝑅)
9 dfrel2 6039 . . . . . 6 (Rel 𝑅𝑅 = 𝑅)
108, 9sylib 220 . . . . 5 (𝑅 ∈ PosetRel → 𝑅 = 𝑅)
1110ineq2d 4187 . . . 4 (𝑅 ∈ PosetRel → (𝑅𝑅) = (𝑅𝑅))
12 incom 4176 . . . 4 (𝑅𝑅) = (𝑅𝑅)
1311, 12syl6eq 2870 . . 3 (𝑅 ∈ PosetRel → (𝑅𝑅) = (𝑅𝑅))
14 psref2 17806 . . 3 (𝑅 ∈ PosetRel → (𝑅𝑅) = ( I ↾ 𝑅))
15 relcnvfld 6124 . . . . 5 (Rel 𝑅 𝑅 = 𝑅)
168, 15syl 17 . . . 4 (𝑅 ∈ PosetRel → 𝑅 = 𝑅)
1716reseq2d 5846 . . 3 (𝑅 ∈ PosetRel → ( I ↾ 𝑅) = ( I ↾ 𝑅))
1813, 14, 173eqtrd 2858 . 2 (𝑅 ∈ PosetRel → (𝑅𝑅) = ( I ↾ 𝑅))
19 cnvexg 7621 . . 3 (𝑅 ∈ PosetRel → 𝑅 ∈ V)
20 isps 17804 . . 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 1336 1 (𝑅 ∈ PosetRel → 𝑅 ∈ PosetRel)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 208   ∧ w3a 1081   = wceq 1530   ∈ wcel 2107  Vcvv 3493   ∩ cin 3933   ⊆ wss 3934  ∪ cuni 4830   I cid 5452  ◡ccnv 5547   ↾ cres 5550   ∘ ccom 5552  Rel wrel 5553  PosetRelcps 17800 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ps 17802 This theorem is referenced by:  cnvpsb  17815  cnvtsr  17824  ordtcnv  21801  xrge0iifhmeo  31167
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