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Theorem dirdm 17281
 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 3809 . . . 4 dom 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅)
2 dmrnssfld 5416 . . . 4 (dom 𝑅 ∪ ran 𝑅) ⊆ 𝑅
31, 2sstri 3645 . . 3 dom 𝑅 𝑅
43a1i 11 . 2 (𝑅 ∈ DirRel → dom 𝑅 𝑅)
5 dmresi 5492 . . 3 dom ( I ↾ 𝑅) = 𝑅
6 eqid 2651 . . . . . . . 8 𝑅 = 𝑅
76isdir 17279 . . . . . . 7 (𝑅 ∈ DirRel → (𝑅 ∈ DirRel ↔ ((Rel 𝑅 ∧ ( I ↾ 𝑅) ⊆ 𝑅) ∧ ((𝑅𝑅) ⊆ 𝑅 ∧ ( 𝑅 × 𝑅) ⊆ (𝑅𝑅)))))
87ibi 256 . . . . . 6 (𝑅 ∈ DirRel → ((Rel 𝑅 ∧ ( I ↾ 𝑅) ⊆ 𝑅) ∧ ((𝑅𝑅) ⊆ 𝑅 ∧ ( 𝑅 × 𝑅) ⊆ (𝑅𝑅))))
98simpld 474 . . . . 5 (𝑅 ∈ DirRel → (Rel 𝑅 ∧ ( I ↾ 𝑅) ⊆ 𝑅))
109simprd 478 . . . 4 (𝑅 ∈ DirRel → ( I ↾ 𝑅) ⊆ 𝑅)
11 dmss 5355 . . . 4 (( I ↾ 𝑅) ⊆ 𝑅 → dom ( I ↾ 𝑅) ⊆ dom 𝑅)
1210, 11syl 17 . . 3 (𝑅 ∈ DirRel → dom ( I ↾ 𝑅) ⊆ dom 𝑅)
135, 12syl5eqssr 3683 . 2 (𝑅 ∈ DirRel → 𝑅 ⊆ dom 𝑅)
144, 13eqssd 3653 1 (𝑅 ∈ DirRel → dom 𝑅 = 𝑅)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1523   ∈ wcel 2030   ∪ cun 3605   ⊆ wss 3607  ∪ cuni 4468   I cid 5052   × cxp 5141  ◡ccnv 5142  dom cdm 5143  ran crn 5144   ↾ cres 5145   ∘ ccom 5147  Rel wrel 5148  DirRelcdir 17275 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-dir 17277 This theorem is referenced by:  dirref  17282  dirge  17284  tailfval  32492  tailf  32495  filnetlem4  32501
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