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Theorem reldir 17838
 Description: A direction is a relation. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 22-Nov-2013.)
Assertion
Ref Expression
reldir (𝑅 ∈ DirRel → Rel 𝑅)

Proof of Theorem reldir
StepHypRef Expression
1 eqid 2798 . . . 4 𝑅 = 𝑅
21isdir 17837 . . 3 (𝑅 ∈ DirRel → (𝑅 ∈ DirRel ↔ ((Rel 𝑅 ∧ ( I ↾ 𝑅) ⊆ 𝑅) ∧ ((𝑅𝑅) ⊆ 𝑅 ∧ ( 𝑅 × 𝑅) ⊆ (𝑅𝑅)))))
32ibi 270 . 2 (𝑅 ∈ DirRel → ((Rel 𝑅 ∧ ( I ↾ 𝑅) ⊆ 𝑅) ∧ ((𝑅𝑅) ⊆ 𝑅 ∧ ( 𝑅 × 𝑅) ⊆ (𝑅𝑅))))
43simplld 767 1 (𝑅 ∈ DirRel → Rel 𝑅)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∈ wcel 2111   ⊆ wss 3881  ∪ cuni 4801   I cid 5425   × cxp 5518  ◡ccnv 5519   ↾ cres 5522   ∘ ccom 5524  Rel wrel 5525  DirRelcdir 17833 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-rab 3115  df-v 3443  df-in 3888  df-ss 3898  df-uni 4802  df-br 5032  df-opab 5094  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-res 5532  df-dir 17835 This theorem is referenced by:  dirtr  17841  dirge  17842
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