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Theorem dirdm 18608

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 4153	. . . 4 ⊢ dom 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅)
2		dmrnssfld 5953	. . . 4 ⊢ (dom 𝑅 ∪ ran 𝑅) ⊆ ∪ ∪ 𝑅
3	1, 2	sstri 3968	. . 3 ⊢ dom 𝑅 ⊆ ∪ ∪ 𝑅
4	3	a1i 11	. 2 ⊢ (𝑅 ∈ DirRel → dom 𝑅 ⊆ ∪ ∪ 𝑅)
5		dmresi 6039	. . 3 ⊢ dom ( I ↾ ∪ ∪ 𝑅) = ∪ ∪ 𝑅
6		eqid 2735	. . . . . . 7 ⊢ ∪ ∪ 𝑅 = ∪ ∪ 𝑅
7	6	isdir 18606	. . . . . 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 769	. . . 4 ⊢ (𝑅 ∈ DirRel → ( I ↾ ∪ ∪ 𝑅) ⊆ 𝑅)
10		dmss 5882	. . . 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 3998	. 2 ⊢ (𝑅 ∈ DirRel → ∪ ∪ 𝑅 ⊆ dom 𝑅)
13	4, 12	eqssd 3976	1 ⊢ (𝑅 ∈ DirRel → dom 𝑅 = ∪ ∪ 𝑅)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∪ cun 3924 ⊆ wss 3926 ∪ cuni 4883 I cid 5547 × cxp 5652 ^◡ccnv 5653 dom cdm 5654 ran crn 5655 ↾ cres 5656 ∘ ccom 5658 Rel wrel 5659 DirRelcdir 18602

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2707 ax-sep 5266 ax-nul 5276 ax-pr 5402

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2714 df-cleq 2727 df-clel 2809 df-ral 3052 df-rex 3061 df-rab 3416 df-v 3461 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-nul 4309 df-if 4501 df-sn 4602 df-pr 4604 df-op 4608 df-uni 4884 df-br 5120 df-opab 5182 df-id 5548 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-dir 18604

This theorem is referenced by: dirref 18609 dirge 18611 tailfval 36336 tailf 36339 filnetlem4 36345

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