Theorem dirdm 18655

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

Step	Hyp	Ref	Expression
1		ssun1 4139	. . . 4 ⊢ dom 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅)
2		dmrnssfld 5965	. . . 4 ⊢ (dom 𝑅 ∪ ran 𝑅) ⊆ ∪ ∪ 𝑅
3	1, 2	sstri 3954	. . 3 ⊢ dom 𝑅 ⊆ ∪ ∪ 𝑅
4	3	a1i 11	. 2 ⊢ (𝑅 ∈ DirRel → dom 𝑅 ⊆ ∪ ∪ 𝑅)
5		dmresi 6055	. . 3 ⊢ dom ( I ↾ ∪ ∪ 𝑅) = ∪ ∪ 𝑅
6		eqid 2769	. . . . . . 7 ⊢ ∪ ∪ 𝑅 = ∪ ∪ 𝑅
7	6	isdir 18653	. . . . . 6 ⊢ (𝑅 ∈ DirRel → (𝑅 ∈ DirRel ↔ ((Rel 𝑅 ∧ ( I ↾ ∪ ∪ 𝑅) ⊆ 𝑅) ∧ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ (∪ ∪ 𝑅 × ∪ ∪ 𝑅) ⊆ (◡𝑅 ∘ 𝑅)))))
8	7	ibi 270	. . . . 5 ⊢ (𝑅 ∈ DirRel → ((Rel 𝑅 ∧ ( I ↾ ∪ ∪ 𝑅) ⊆ 𝑅) ∧ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ (∪ ∪ 𝑅 × ∪ ∪ 𝑅) ⊆ (◡𝑅 ∘ 𝑅))))
9	8	simplrd 781	. . . 4 ⊢ (𝑅 ∈ DirRel → ( I ↾ ∪ ∪ 𝑅) ⊆ 𝑅)
10		dmss 5893	. . . 4 ⊢ (( I ↾ ∪ ∪ 𝑅) ⊆ 𝑅 → dom ( I ↾ ∪ ∪ 𝑅) ⊆ dom 𝑅)
11	9, 10	syl 18	. . 3 ⊢ (𝑅 ∈ DirRel → dom ( I ↾ ∪ ∪ 𝑅) ⊆ dom 𝑅)
12	5, 11	eqsstrrid 3984	. 2 ⊢ (𝑅 ∈ DirRel → ∪ ∪ 𝑅 ⊆ dom 𝑅)
13	4, 12	eqssd 3962	1 ⊢ (𝑅 ∈ DirRel → dom 𝑅 = ∪ ∪ 𝑅)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149   ∪ cun 3911   ⊆ wss 3913  ∪ cuni 4876   I cid 5556   × cxp 5660  ^◡ccnv 5661  dom cdm 5662  ran crn 5663   ↾ cres 5664   ∘ ccom 5666  Rel wrel 5667  DirRelcdir 18649

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5261  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-dir 18651

This theorem is referenced by:  dirref  18656  dirge  18658  tailfval  36771  tailf  36774  filnetlem4  36780