Theorem cnvpsb 18544

Description: The converse of a poset is a poset. (Contributed by FL, 5-Jan-2009.)

Assertion

Ref	Expression
cnvpsb	⊢ (Rel 𝑅 → (𝑅 ∈ PosetRel ↔ ◡𝑅 ∈ PosetRel))

Proof of Theorem cnvpsb

Step	Hyp	Ref	Expression
1		cnvps 18543	. 2 ⊢ (𝑅 ∈ PosetRel → ◡𝑅 ∈ PosetRel)
2		cnvps 18543	. . 3 ⊢ (◡𝑅 ∈ PosetRel → ◡◡𝑅 ∈ PosetRel)
3		dfrel2 6164	. . . 4 ⊢ (Rel 𝑅 ↔ ◡◡𝑅 = 𝑅)
4		eleq1 2817	. . . . 5 ⊢ (◡◡𝑅 = 𝑅 → (◡◡𝑅 ∈ PosetRel ↔ 𝑅 ∈ PosetRel))
5	4	biimpd 229	. . . 4 ⊢ (◡◡𝑅 = 𝑅 → (◡◡𝑅 ∈ PosetRel → 𝑅 ∈ PosetRel))
6	3, 5	sylbi 217	. . 3 ⊢ (Rel 𝑅 → (◡◡𝑅 ∈ PosetRel → 𝑅 ∈ PosetRel))
7	2, 6	syl5 34	. 2 ⊢ (Rel 𝑅 → (◡𝑅 ∈ PosetRel → 𝑅 ∈ PosetRel))
8	1, 7	impbid2 226	1 ⊢ (Rel 𝑅 → (𝑅 ∈ PosetRel ↔ ◡𝑅 ∈ PosetRel))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1540   ∈ wcel 2109  ^◡ccnv 5639  Rel wrel 5645  PosetRelcps 18529

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 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-sep 5253  ax-nul 5263  ax-pow 5322  ax-pr 5389  ax-un 7713

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-nul 4299  df-if 4491  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5110  df-opab 5172  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ps 18531

This theorem is referenced by: (None)