MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  reldir Structured version   Visualization version   GIF version

Theorem reldir 17454
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 2760 . . . 4 𝑅 = 𝑅
21isdir 17453 . . 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 808 1 (𝑅 ∈ DirRel → Rel 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wcel 2139  wss 3715   cuni 4588   I cid 5173   × cxp 5264  ccnv 5265  cres 5268  ccom 5270  Rel wrel 5271  DirRelcdir 17449
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-rex 3056  df-v 3342  df-in 3722  df-ss 3729  df-uni 4589  df-br 4805  df-opab 4865  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-res 5278  df-dir 17451
This theorem is referenced by:  dirtr  17457  dirge  17458
  Copyright terms: Public domain W3C validator