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Theorem dirref 17839
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 17838 . . . . . 6 (𝑅 ∈ DirRel → dom 𝑅 = 𝑅)
31, 2syl5eq 2868 . . . . 5 (𝑅 ∈ DirRel → 𝑋 = 𝑅)
43reseq2d 5847 . . . 4 (𝑅 ∈ DirRel → ( I ↾ 𝑋) = ( I ↾ 𝑅))
5 eqid 2821 . . . . . . 7 𝑅 = 𝑅
65isdir 17836 . . . . . 6 (𝑅 ∈ DirRel → (𝑅 ∈ DirRel ↔ ((Rel 𝑅 ∧ ( I ↾ 𝑅) ⊆ 𝑅) ∧ ((𝑅𝑅) ⊆ 𝑅 ∧ ( 𝑅 × 𝑅) ⊆ (𝑅𝑅)))))
76ibi 269 . . . . 5 (𝑅 ∈ DirRel → ((Rel 𝑅 ∧ ( I ↾ 𝑅) ⊆ 𝑅) ∧ ((𝑅𝑅) ⊆ 𝑅 ∧ ( 𝑅 × 𝑅) ⊆ (𝑅𝑅))))
87simplrd 768 . . . 4 (𝑅 ∈ DirRel → ( I ↾ 𝑅) ⊆ 𝑅)
94, 8eqsstrd 4004 . . 3 (𝑅 ∈ DirRel → ( I ↾ 𝑋) ⊆ 𝑅)
109ssbrd 5101 . 2 (𝑅 ∈ DirRel → (𝐴( I ↾ 𝑋)𝐴𝐴𝑅𝐴))
11 eqid 2821 . . 3 𝐴 = 𝐴
12 resieq 5858 . . . 4 ((𝐴𝑋𝐴𝑋) → (𝐴( I ↾ 𝑋)𝐴𝐴 = 𝐴))
1312anidms 569 . . 3 (𝐴𝑋 → (𝐴( I ↾ 𝑋)𝐴𝐴 = 𝐴))
1411, 13mpbiri 260 . 2 (𝐴𝑋𝐴( I ↾ 𝑋)𝐴)
1510, 14impel 508 1 ((𝑅 ∈ DirRel ∧ 𝐴𝑋) → 𝐴𝑅𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398   = wceq 1533  wcel 2110  wss 3935   cuni 4831   class class class wbr 5058   I cid 5453   × cxp 5547  ccnv 5548  dom cdm 5549  cres 5551  ccom 5553  Rel wrel 5554  DirRelcdir 17832
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5059  df-opab 5121  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-dir 17834
This theorem is referenced by:  tailini  33719
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