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Theorem dirref 17501
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 2765 . . . 4 𝐴 = 𝐴
2 resieq 5583 . . . . 5 ((𝐴𝑋𝐴𝑋) → (𝐴( I ↾ 𝑋)𝐴𝐴 = 𝐴))
32anidms 562 . . . 4 (𝐴𝑋 → (𝐴( I ↾ 𝑋)𝐴𝐴 = 𝐴))
41, 3mpbiri 249 . . 3 (𝐴𝑋𝐴( I ↾ 𝑋)𝐴)
5 dirref.1 . . . . . . 7 𝑋 = dom 𝑅
6 dirdm 17500 . . . . . . 7 (𝑅 ∈ DirRel → dom 𝑅 = 𝑅)
75, 6syl5eq 2811 . . . . . 6 (𝑅 ∈ DirRel → 𝑋 = 𝑅)
87reseq2d 5565 . . . . 5 (𝑅 ∈ DirRel → ( I ↾ 𝑋) = ( I ↾ 𝑅))
9 eqid 2765 . . . . . . . . 9 𝑅 = 𝑅
109isdir 17498 . . . . . . . 8 (𝑅 ∈ DirRel → (𝑅 ∈ DirRel ↔ ((Rel 𝑅 ∧ ( I ↾ 𝑅) ⊆ 𝑅) ∧ ((𝑅𝑅) ⊆ 𝑅 ∧ ( 𝑅 × 𝑅) ⊆ (𝑅𝑅)))))
1110ibi 258 . . . . . . 7 (𝑅 ∈ DirRel → ((Rel 𝑅 ∧ ( I ↾ 𝑅) ⊆ 𝑅) ∧ ((𝑅𝑅) ⊆ 𝑅 ∧ ( 𝑅 × 𝑅) ⊆ (𝑅𝑅))))
1211simpld 488 . . . . . 6 (𝑅 ∈ DirRel → (Rel 𝑅 ∧ ( I ↾ 𝑅) ⊆ 𝑅))
1312simprd 489 . . . . 5 (𝑅 ∈ DirRel → ( I ↾ 𝑅) ⊆ 𝑅)
148, 13eqsstrd 3799 . . . 4 (𝑅 ∈ DirRel → ( I ↾ 𝑋) ⊆ 𝑅)
1514ssbrd 4852 . . 3 (𝑅 ∈ DirRel → (𝐴( I ↾ 𝑋)𝐴𝐴𝑅𝐴))
164, 15syl5 34 . 2 (𝑅 ∈ DirRel → (𝐴𝑋𝐴𝑅𝐴))
1716imp 395 1 ((𝑅 ∈ DirRel ∧ 𝐴𝑋) → 𝐴𝑅𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1652  wcel 2155  wss 3732   cuni 4594   class class class wbr 4809   I cid 5184   × cxp 5275  ccnv 5276  dom cdm 5277  cres 5279  ccom 5281  Rel wrel 5282  DirRelcdir 17494
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pr 5062
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-br 4810  df-opab 4872  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-dir 17496
This theorem is referenced by:  tailini  32814
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