Theorem dirdm 18786

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 4178	. . . 4 ⊢ dom 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅)
2		dmrnssfld 5960	. . . 4 ⊢ (dom 𝑅 ∪ ran 𝑅) ⊆ ∪ ∪ 𝑅
3	1, 2	sstri 3921	. . 3 ⊢ dom 𝑅 ⊆ ∪ ∪ 𝑅
4	3	a1i 11	. 2 ⊢ (𝑅 ∈ DirRel → dom 𝑅 ⊆ ∪ ∪ 𝑅)
5		dmresi 6049	. . 3 ⊢ dom ( I ↾ ∪ ∪ 𝑅) = ∪ ∪ 𝑅
6		eqid 2763	. . . . . . 7 ⊢ ∪ ∪ 𝑅 = ∪ ∪ 𝑅
7	6	isdir 18784	. . . . . 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 782	. . . 4 ⊢ (𝑅 ∈ DirRel → ( I ↾ ∪ ∪ 𝑅) ⊆ 𝑅)
10		dmss 5888	. . . 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 3951	. 2 ⊢ (𝑅 ∈ DirRel → ∪ ∪ 𝑅 ⊆ dom 𝑅)
13	4, 12	eqssd 3929	1 ⊢ (𝑅 ∈ DirRel → dom 𝑅 = ∪ ∪ 𝑅)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2146   ⊆ wss 3899   ∪ cun 4151  ∪ cuni 4869   I cid 5548   × cxp 5652  ^◡ccnv 5653  dom cdm 5654  ran crn 5655   ↾ cres 5656   ∘ ccom 5658  Rel wrel 5659  DirRelcdir 18780

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735  ax-sep 5251  ax-pr 5394

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3416  df-v 3455  df-ss 3900  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-sn 4588  df-pr 4590  df-op 4838  df-uni 4870  df-br 5106  df-opab 5170  df-id 5549  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-dir 18782

This theorem is used by:  dirref  18787  dirge  18789  tailfval  37287  tailf  37290  filnetlem4  37296