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Theorem reldir 18105
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 2737 . . . 4 𝑅 = 𝑅
21isdir 18104 . . 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 768 1 (𝑅 ∈ DirRel → Rel 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wcel 2110  wss 3866   cuni 4819   I cid 5454   × cxp 5549  ccnv 5550  cres 5553  ccom 5555  Rel wrel 5556  DirRelcdir 18100
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708
This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1546  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-rab 3070  df-v 3410  df-in 3873  df-ss 3883  df-uni 4820  df-br 5054  df-opab 5116  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-res 5563  df-dir 18102
This theorem is referenced by:  dirtr  18108  dirge  18109
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