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Theorem dirdm 18535

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 4132	. . . 4 ⊢ dom 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅)
2		dmrnssfld 5931	. . . 4 ⊢ (dom 𝑅 ∪ ran 𝑅) ⊆ ∪ ∪ 𝑅
3	1, 2	sstri 3945	. . 3 ⊢ dom 𝑅 ⊆ ∪ ∪ 𝑅
4	3	a1i 11	. 2 ⊢ (𝑅 ∈ DirRel → dom 𝑅 ⊆ ∪ ∪ 𝑅)
5		dmresi 6019	. . 3 ⊢ dom ( I ↾ ∪ ∪ 𝑅) = ∪ ∪ 𝑅
6		eqid 2737	. . . . . . 7 ⊢ ∪ ∪ 𝑅 = ∪ ∪ 𝑅
7	6	isdir 18533	. . . . . 6 ⊢ (𝑅 ∈ DirRel → (𝑅 ∈ DirRel ↔ ((Rel 𝑅 ∧ ( I ↾ ∪ ∪ 𝑅) ⊆ 𝑅) ∧ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ (∪ ∪ 𝑅 × ∪ ∪ 𝑅) ⊆ (^◡𝑅 ∘ 𝑅)))))
8	7	ibi 267	. . . . 5 ⊢ (𝑅 ∈ DirRel → ((Rel 𝑅 ∧ ( I ↾ ∪ ∪ 𝑅) ⊆ 𝑅) ∧ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ (∪ ∪ 𝑅 × ∪ ∪ 𝑅) ⊆ (^◡𝑅 ∘ 𝑅))))
9	8	simplrd 770	. . . 4 ⊢ (𝑅 ∈ DirRel → ( I ↾ ∪ ∪ 𝑅) ⊆ 𝑅)
10		dmss 5859	. . . 4 ⊢ (( I ↾ ∪ ∪ 𝑅) ⊆ 𝑅 → dom ( I ↾ ∪ ∪ 𝑅) ⊆ dom 𝑅)
11	9, 10	syl 17	. . 3 ⊢ (𝑅 ∈ DirRel → dom ( I ↾ ∪ ∪ 𝑅) ⊆ dom 𝑅)
12	5, 11	eqsstrrid 3975	. 2 ⊢ (𝑅 ∈ DirRel → ∪ ∪ 𝑅 ⊆ dom 𝑅)
13	4, 12	eqssd 3953	1 ⊢ (𝑅 ∈ DirRel → dom 𝑅 = ∪ ∪ 𝑅)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ∪ cun 3901 ⊆ wss 3903 ∪ cuni 4865 I cid 5526 × cxp 5630 ^◡ccnv 5631 dom cdm 5632 ran crn 5633 ↾ cres 5634 ∘ ccom 5636 Rel wrel 5637 DirRelcdir 18529

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-ext 2709 ax-sep 5243 ax-pr 5379

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-sb 2069 df-clab 2716 df-cleq 2729 df-clel 2812 df-ral 3053 df-rex 3063 df-rab 3402 df-v 3444 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-nul 4288 df-if 4482 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-br 5101 df-opab 5163 df-id 5527 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-dir 18531

This theorem is referenced by: dirref 18536 dirge 18538 tailfval 36588 tailf 36591 filnetlem4 36597

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