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Theorem reldir 17154
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 2621 . . . 4 𝑅 = 𝑅
21isdir 17153 . . 3 (𝑅 ∈ DirRel → (𝑅 ∈ DirRel ↔ ((Rel 𝑅 ∧ ( I ↾ 𝑅) ⊆ 𝑅) ∧ ((𝑅𝑅) ⊆ 𝑅 ∧ ( 𝑅 × 𝑅) ⊆ (𝑅𝑅)))))
32ibi 256 . 2 (𝑅 ∈ DirRel → ((Rel 𝑅 ∧ ( I ↾ 𝑅) ⊆ 𝑅) ∧ ((𝑅𝑅) ⊆ 𝑅 ∧ ( 𝑅 × 𝑅) ⊆ (𝑅𝑅))))
43simplld 790 1 (𝑅 ∈ DirRel → Rel 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wcel 1987  wss 3555   cuni 4402   I cid 4984   × cxp 5072  ccnv 5073  cres 5076  ccom 5078  Rel wrel 5079  DirRelcdir 17149
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rex 2913  df-v 3188  df-in 3562  df-ss 3569  df-uni 4403  df-br 4614  df-opab 4674  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-res 5086  df-dir 17151
This theorem is referenced by:  dirtr  17157  dirge  17158
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