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Theorem dirref 18360
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 18359 . . . . . 6 (𝑅 ∈ DirRel → dom 𝑅 = 𝑅)
31, 2eqtrid 2788 . . . . 5 (𝑅 ∈ DirRel → 𝑋 = 𝑅)
43reseq2d 5899 . . . 4 (𝑅 ∈ DirRel → ( I ↾ 𝑋) = ( I ↾ 𝑅))
5 eqid 2736 . . . . . . 7 𝑅 = 𝑅
65isdir 18357 . . . . . 6 (𝑅 ∈ DirRel → (𝑅 ∈ DirRel ↔ ((Rel 𝑅 ∧ ( I ↾ 𝑅) ⊆ 𝑅) ∧ ((𝑅𝑅) ⊆ 𝑅 ∧ ( 𝑅 × 𝑅) ⊆ (𝑅𝑅)))))
76ibi 268 . . . . 5 (𝑅 ∈ DirRel → ((Rel 𝑅 ∧ ( I ↾ 𝑅) ⊆ 𝑅) ∧ ((𝑅𝑅) ⊆ 𝑅 ∧ ( 𝑅 × 𝑅) ⊆ (𝑅𝑅))))
87simplrd 768 . . . 4 (𝑅 ∈ DirRel → ( I ↾ 𝑅) ⊆ 𝑅)
94, 8eqsstrd 3964 . . 3 (𝑅 ∈ DirRel → ( I ↾ 𝑋) ⊆ 𝑅)
109ssbrd 5124 . 2 (𝑅 ∈ DirRel → (𝐴( I ↾ 𝑋)𝐴𝐴𝑅𝐴))
11 eqid 2736 . . 3 𝐴 = 𝐴
12 resieq 5910 . . . 4 ((𝐴𝑋𝐴𝑋) → (𝐴( I ↾ 𝑋)𝐴𝐴 = 𝐴))
1312anidms 568 . . 3 (𝐴𝑋 → (𝐴( I ↾ 𝑋)𝐴𝐴 = 𝐴))
1411, 13mpbiri 259 . 2 (𝐴𝑋𝐴( I ↾ 𝑋)𝐴)
1510, 14impel 507 1 ((𝑅 ∈ DirRel ∧ 𝐴𝑋) → 𝐴𝑅𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1539  wcel 2104  wss 3892   cuni 4844   class class class wbr 5081   I cid 5495   × cxp 5594  ccnv 5595  dom cdm 5596  cres 5598  ccom 5600  Rel wrel 5601  DirRelcdir 18353
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2707  ax-sep 5232  ax-nul 5239  ax-pr 5361
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3063  df-rex 3072  df-rab 3287  df-v 3439  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-br 5082  df-opab 5144  df-id 5496  df-xp 5602  df-rel 5603  df-cnv 5604  df-co 5605  df-dm 5606  df-rn 5607  df-res 5608  df-dir 18355
This theorem is referenced by:  tailini  34606
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