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Theorem dirdm 18443
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 4130 . . . 4 dom 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅)
2 dmrnssfld 5923 . . . 4 (dom 𝑅 ∪ ran 𝑅) ⊆ 𝑅
31, 2sstri 3951 . . 3 dom 𝑅 𝑅
43a1i 11 . 2 (𝑅 ∈ DirRel → dom 𝑅 𝑅)
5 dmresi 6003 . . 3 dom ( I ↾ 𝑅) = 𝑅
6 eqid 2736 . . . . . . 7 𝑅 = 𝑅
76isdir 18441 . . . . . 6 (𝑅 ∈ DirRel → (𝑅 ∈ DirRel ↔ ((Rel 𝑅 ∧ ( I ↾ 𝑅) ⊆ 𝑅) ∧ ((𝑅𝑅) ⊆ 𝑅 ∧ ( 𝑅 × 𝑅) ⊆ (𝑅𝑅)))))
87ibi 266 . . . . 5 (𝑅 ∈ DirRel → ((Rel 𝑅 ∧ ( I ↾ 𝑅) ⊆ 𝑅) ∧ ((𝑅𝑅) ⊆ 𝑅 ∧ ( 𝑅 × 𝑅) ⊆ (𝑅𝑅))))
98simplrd 768 . . . 4 (𝑅 ∈ DirRel → ( I ↾ 𝑅) ⊆ 𝑅)
10 dmss 5856 . . . 4 (( I ↾ 𝑅) ⊆ 𝑅 → dom ( I ↾ 𝑅) ⊆ dom 𝑅)
119, 10syl 17 . . 3 (𝑅 ∈ DirRel → dom ( I ↾ 𝑅) ⊆ dom 𝑅)
125, 11eqsstrrid 3991 . 2 (𝑅 ∈ DirRel → 𝑅 ⊆ dom 𝑅)
134, 12eqssd 3959 1 (𝑅 ∈ DirRel → dom 𝑅 = 𝑅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  cun 3906  wss 3908   cuni 4863   I cid 5528   × cxp 5629  ccnv 5630  dom cdm 5631  ran crn 5632  cres 5633  ccom 5635  Rel wrel 5636  DirRelcdir 18437
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707  ax-sep 5254  ax-nul 5261  ax-pr 5382
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-br 5104  df-opab 5166  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-dir 18439
This theorem is referenced by:  dirref  18444  dirge  18446  tailfval  34776  tailf  34779  filnetlem4  34785
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