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Theorem dirref 18658
Description: A direction is reflexive. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 22-Nov-2013.)
Hypothesis
Ref Expression
dirref.1 𝑋 = dom 𝑅
Assertion
Ref Expression
dirref ((𝑅 ∈ DirRel ∧ 𝐴𝑋) → 𝐴𝑅𝐴)

Proof of Theorem dirref
StepHypRef Expression
1 dirref.1 . . . . . 6 𝑋 = dom 𝑅
2 dirdm 18657 . . . . . 6 (𝑅 ∈ DirRel → dom 𝑅 = 𝑅)
31, 2eqtrid 2786 . . . . 5 (𝑅 ∈ DirRel → 𝑋 = 𝑅)
43reseq2d 5999 . . . 4 (𝑅 ∈ DirRel → ( I ↾ 𝑋) = ( I ↾ 𝑅))
5 eqid 2734 . . . . . . 7 𝑅 = 𝑅
65isdir 18655 . . . . . 6 (𝑅 ∈ DirRel → (𝑅 ∈ DirRel ↔ ((Rel 𝑅 ∧ ( I ↾ 𝑅) ⊆ 𝑅) ∧ ((𝑅𝑅) ⊆ 𝑅 ∧ ( 𝑅 × 𝑅) ⊆ (𝑅𝑅)))))
76ibi 267 . . . . 5 (𝑅 ∈ DirRel → ((Rel 𝑅 ∧ ( I ↾ 𝑅) ⊆ 𝑅) ∧ ((𝑅𝑅) ⊆ 𝑅 ∧ ( 𝑅 × 𝑅) ⊆ (𝑅𝑅))))
87simplrd 770 . . . 4 (𝑅 ∈ DirRel → ( I ↾ 𝑅) ⊆ 𝑅)
94, 8eqsstrd 4033 . . 3 (𝑅 ∈ DirRel → ( I ↾ 𝑋) ⊆ 𝑅)
109ssbrd 5190 . 2 (𝑅 ∈ DirRel → (𝐴( I ↾ 𝑋)𝐴𝐴𝑅𝐴))
11 eqid 2734 . . 3 𝐴 = 𝐴
12 resieq 6010 . . . 4 ((𝐴𝑋𝐴𝑋) → (𝐴( I ↾ 𝑋)𝐴𝐴 = 𝐴))
1312anidms 566 . . 3 (𝐴𝑋 → (𝐴( I ↾ 𝑋)𝐴𝐴 = 𝐴))
1411, 13mpbiri 258 . 2 (𝐴𝑋𝐴( I ↾ 𝑋)𝐴)
1510, 14impel 505 1 ((𝑅 ∈ DirRel ∧ 𝐴𝑋) → 𝐴𝑅𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1536  wcel 2105  wss 3962   cuni 4911   class class class wbr 5147   I cid 5581   × cxp 5686  ccnv 5687  dom cdm 5688  cres 5690  ccom 5692  Rel wrel 5693  DirRelcdir 18651
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-dir 18653
This theorem is referenced by:  tailini  36358
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