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Theorem dirref 17436
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 2760 . . . 4 𝐴 = 𝐴
2 resieq 5565 . . . . 5 ((𝐴𝑋𝐴𝑋) → (𝐴( I ↾ 𝑋)𝐴𝐴 = 𝐴))
32anidms 680 . . . 4 (𝐴𝑋 → (𝐴( I ↾ 𝑋)𝐴𝐴 = 𝐴))
41, 3mpbiri 248 . . 3 (𝐴𝑋𝐴( I ↾ 𝑋)𝐴)
5 dirref.1 . . . . . . 7 𝑋 = dom 𝑅
6 dirdm 17435 . . . . . . 7 (𝑅 ∈ DirRel → dom 𝑅 = 𝑅)
75, 6syl5eq 2806 . . . . . 6 (𝑅 ∈ DirRel → 𝑋 = 𝑅)
87reseq2d 5551 . . . . 5 (𝑅 ∈ DirRel → ( I ↾ 𝑋) = ( I ↾ 𝑅))
9 eqid 2760 . . . . . . . . 9 𝑅 = 𝑅
109isdir 17433 . . . . . . . 8 (𝑅 ∈ DirRel → (𝑅 ∈ DirRel ↔ ((Rel 𝑅 ∧ ( I ↾ 𝑅) ⊆ 𝑅) ∧ ((𝑅𝑅) ⊆ 𝑅 ∧ ( 𝑅 × 𝑅) ⊆ (𝑅𝑅)))))
1110ibi 256 . . . . . . 7 (𝑅 ∈ DirRel → ((Rel 𝑅 ∧ ( I ↾ 𝑅) ⊆ 𝑅) ∧ ((𝑅𝑅) ⊆ 𝑅 ∧ ( 𝑅 × 𝑅) ⊆ (𝑅𝑅))))
1211simpld 477 . . . . . 6 (𝑅 ∈ DirRel → (Rel 𝑅 ∧ ( I ↾ 𝑅) ⊆ 𝑅))
1312simprd 482 . . . . 5 (𝑅 ∈ DirRel → ( I ↾ 𝑅) ⊆ 𝑅)
148, 13eqsstrd 3780 . . . 4 (𝑅 ∈ DirRel → ( I ↾ 𝑋) ⊆ 𝑅)
1514ssbrd 4847 . . 3 (𝑅 ∈ DirRel → (𝐴( I ↾ 𝑋)𝐴𝐴𝑅𝐴))
164, 15syl5 34 . 2 (𝑅 ∈ DirRel → (𝐴𝑋𝐴𝑅𝐴))
1716imp 444 1 ((𝑅 ∈ DirRel ∧ 𝐴𝑋) → 𝐴𝑅𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  wss 3715   cuni 4588   class class class wbr 4804   I cid 5173   × cxp 5264  ccnv 5265  dom cdm 5266  cres 5268  ccom 5270  Rel wrel 5271  DirRelcdir 17429
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-dir 17431
This theorem is referenced by:  tailini  32677
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