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Theorem dirref 17058
 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 eqid 2610 . . . 4 𝐴 = 𝐴
2 resieq 5327 . . . . 5 ((𝐴𝑋𝐴𝑋) → (𝐴( I ↾ 𝑋)𝐴𝐴 = 𝐴))
32anidms 675 . . . 4 (𝐴𝑋 → (𝐴( I ↾ 𝑋)𝐴𝐴 = 𝐴))
41, 3mpbiri 247 . . 3 (𝐴𝑋𝐴( I ↾ 𝑋)𝐴)
5 dirref.1 . . . . . . 7 𝑋 = dom 𝑅
6 dirdm 17057 . . . . . . 7 (𝑅 ∈ DirRel → dom 𝑅 = 𝑅)
75, 6syl5eq 2656 . . . . . 6 (𝑅 ∈ DirRel → 𝑋 = 𝑅)
87reseq2d 5317 . . . . 5 (𝑅 ∈ DirRel → ( I ↾ 𝑋) = ( I ↾ 𝑅))
9 eqid 2610 . . . . . . . . 9 𝑅 = 𝑅
109isdir 17055 . . . . . . . 8 (𝑅 ∈ DirRel → (𝑅 ∈ DirRel ↔ ((Rel 𝑅 ∧ ( I ↾ 𝑅) ⊆ 𝑅) ∧ ((𝑅𝑅) ⊆ 𝑅 ∧ ( 𝑅 × 𝑅) ⊆ (𝑅𝑅)))))
1110ibi 255 . . . . . . 7 (𝑅 ∈ DirRel → ((Rel 𝑅 ∧ ( I ↾ 𝑅) ⊆ 𝑅) ∧ ((𝑅𝑅) ⊆ 𝑅 ∧ ( 𝑅 × 𝑅) ⊆ (𝑅𝑅))))
1211simpld 474 . . . . . 6 (𝑅 ∈ DirRel → (Rel 𝑅 ∧ ( I ↾ 𝑅) ⊆ 𝑅))
1312simprd 478 . . . . 5 (𝑅 ∈ DirRel → ( I ↾ 𝑅) ⊆ 𝑅)
148, 13eqsstrd 3602 . . . 4 (𝑅 ∈ DirRel → ( I ↾ 𝑋) ⊆ 𝑅)
1514ssbrd 4626 . . 3 (𝑅 ∈ DirRel → (𝐴( I ↾ 𝑋)𝐴𝐴𝑅𝐴))
164, 15syl5 33 . 2 (𝑅 ∈ DirRel → (𝐴𝑋𝐴𝑅𝐴))
1716imp 444 1 ((𝑅 ∈ DirRel ∧ 𝐴𝑋) → 𝐴𝑅𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ⊆ wss 3540  ∪ cuni 4372   class class class wbr 4583   I cid 4948   × cxp 5036  ◡ccnv 5037  dom cdm 5038   ↾ cres 5040   ∘ ccom 5042  Rel wrel 5043  DirRelcdir 17051 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-dir 17053 This theorem is referenced by:  tailini  31541
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