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Theorem dirdm 18645
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 4178 . . . 4 dom 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅)
2 dmrnssfld 5984 . . . 4 (dom 𝑅 ∪ ran 𝑅) ⊆ 𝑅
31, 2sstri 3993 . . 3 dom 𝑅 𝑅
43a1i 11 . 2 (𝑅 ∈ DirRel → dom 𝑅 𝑅)
5 dmresi 6070 . . 3 dom ( I ↾ 𝑅) = 𝑅
6 eqid 2737 . . . . . . 7 𝑅 = 𝑅
76isdir 18643 . . . . . 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 5913 . . . 4 (( I ↾ 𝑅) ⊆ 𝑅 → dom ( I ↾ 𝑅) ⊆ dom 𝑅)
119, 10syl 17 . . 3 (𝑅 ∈ DirRel → dom ( I ↾ 𝑅) ⊆ dom 𝑅)
125, 11eqsstrrid 4023 . 2 (𝑅 ∈ DirRel → 𝑅 ⊆ dom 𝑅)
134, 12eqssd 4001 1 (𝑅 ∈ DirRel → dom 𝑅 = 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  cun 3949  wss 3951   cuni 4907   I cid 5577   × cxp 5683  ccnv 5684  dom cdm 5685  ran crn 5686  cres 5687  ccom 5689  Rel wrel 5690  DirRelcdir 18639
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-dir 18641
This theorem is referenced by:  dirref  18646  dirge  18648  tailfval  36373  tailf  36376  filnetlem4  36382
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