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Theorem dirdm 18657
Description: A direction's domain is equal to its field. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 22-Nov-2013.)
Assertion
Ref Expression
dirdm (𝑅 ∈ DirRel → dom 𝑅 = 𝑅)

Proof of Theorem dirdm
StepHypRef Expression
1 ssun1 4187 . . . 4 dom 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅)
2 dmrnssfld 5986 . . . 4 (dom 𝑅 ∪ ran 𝑅) ⊆ 𝑅
31, 2sstri 4004 . . 3 dom 𝑅 𝑅
43a1i 11 . 2 (𝑅 ∈ DirRel → dom 𝑅 𝑅)
5 dmresi 6071 . . 3 dom ( I ↾ 𝑅) = 𝑅
6 eqid 2734 . . . . . . 7 𝑅 = 𝑅
76isdir 18655 . . . . . 6 (𝑅 ∈ DirRel → (𝑅 ∈ DirRel ↔ ((Rel 𝑅 ∧ ( I ↾ 𝑅) ⊆ 𝑅) ∧ ((𝑅𝑅) ⊆ 𝑅 ∧ ( 𝑅 × 𝑅) ⊆ (𝑅𝑅)))))
87ibi 267 . . . . 5 (𝑅 ∈ DirRel → ((Rel 𝑅 ∧ ( I ↾ 𝑅) ⊆ 𝑅) ∧ ((𝑅𝑅) ⊆ 𝑅 ∧ ( 𝑅 × 𝑅) ⊆ (𝑅𝑅))))
98simplrd 770 . . . 4 (𝑅 ∈ DirRel → ( I ↾ 𝑅) ⊆ 𝑅)
10 dmss 5915 . . . 4 (( I ↾ 𝑅) ⊆ 𝑅 → dom ( I ↾ 𝑅) ⊆ dom 𝑅)
119, 10syl 17 . . 3 (𝑅 ∈ DirRel → dom ( I ↾ 𝑅) ⊆ dom 𝑅)
125, 11eqsstrrid 4044 . 2 (𝑅 ∈ DirRel → 𝑅 ⊆ dom 𝑅)
134, 12eqssd 4012 1 (𝑅 ∈ DirRel → dom 𝑅 = 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1536  wcel 2105  cun 3960  wss 3962   cuni 4911   I cid 5581   × cxp 5686  ccnv 5687  dom cdm 5688  ran crn 5689  cres 5690  ccom 5692  Rel wrel 5693  DirRelcdir 18651
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-dir 18653
This theorem is referenced by:  dirref  18658  dirge  18660  tailfval  36354  tailf  36357  filnetlem4  36363
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