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Theorem dirref 18554
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 18553 . . . . . 6 (𝑅 ∈ DirRel → dom 𝑅 = 𝑅)
31, 2eqtrid 2785 . . . . 5 (𝑅 ∈ DirRel → 𝑋 = 𝑅)
43reseq2d 5982 . . . 4 (𝑅 ∈ DirRel → ( I ↾ 𝑋) = ( I ↾ 𝑅))
5 eqid 2733 . . . . . . 7 𝑅 = 𝑅
65isdir 18551 . . . . . 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 769 . . . 4 (𝑅 ∈ DirRel → ( I ↾ 𝑅) ⊆ 𝑅)
94, 8eqsstrd 4021 . . 3 (𝑅 ∈ DirRel → ( I ↾ 𝑋) ⊆ 𝑅)
109ssbrd 5192 . 2 (𝑅 ∈ DirRel → (𝐴( I ↾ 𝑋)𝐴𝐴𝑅𝐴))
11 eqid 2733 . . 3 𝐴 = 𝐴
12 resieq 5993 . . . 4 ((𝐴𝑋𝐴𝑋) → (𝐴( I ↾ 𝑋)𝐴𝐴 = 𝐴))
1312anidms 568 . . 3 (𝐴𝑋 → (𝐴( I ↾ 𝑋)𝐴𝐴 = 𝐴))
1411, 13mpbiri 258 . 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 1542  wcel 2107  wss 3949   cuni 4909   class class class wbr 5149   I cid 5574   × cxp 5675  ccnv 5676  dom cdm 5677  cres 5679  ccom 5681  Rel wrel 5682  DirRelcdir 18547
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-dir 18549
This theorem is referenced by:  tailini  35261
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