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Theorem dirref 18689

Description: A direction is reflexive. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 22-Nov-2013.)

**Hypothesis**
Ref	Expression
dirref.1	⊢ 𝑋 = dom 𝑅

**Assertion**
Ref	Expression
dirref	⊢ ((𝑅 ∈ DirRel ∧ 𝐴 ∈ 𝑋) → 𝐴𝑅𝐴)

**Proof of Theorem dirref**
Step	Hyp	Ref	Expression
1		dirref.1	. . . . . 6 ⊢ 𝑋 = dom 𝑅
2		dirdm 18688	. . . . . 6 ⊢ (𝑅 ∈ DirRel → dom 𝑅 = ∪ ∪ 𝑅)
3	1, 2	eqtrid 2807	. . . . 5 ⊢ (𝑅 ∈ DirRel → 𝑋 = ∪ ∪ 𝑅)
4	3	reseq2d 5972	. . . 4 ⊢ (𝑅 ∈ DirRel → ( I ↾ 𝑋) = ( I ↾ ∪ ∪ 𝑅))
5		eqid 2760	. . . . . . 7 ⊢ ∪ ∪ 𝑅 = ∪ ∪ 𝑅
6	5	isdir 18686	. . . . . 6 ⊢ (𝑅 ∈ DirRel → (𝑅 ∈ DirRel ↔ ((Rel 𝑅 ∧ ( I ↾ ∪ ∪ 𝑅) ⊆ 𝑅) ∧ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ (∪ ∪ 𝑅 × ∪ ∪ 𝑅) ⊆ (^◡𝑅 ∘ 𝑅)))))
7	6	ibi 270	. . . . 5 ⊢ (𝑅 ∈ DirRel → ((Rel 𝑅 ∧ ( I ↾ ∪ ∪ 𝑅) ⊆ 𝑅) ∧ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ (∪ ∪ 𝑅 × ∪ ∪ 𝑅) ⊆ (^◡𝑅 ∘ 𝑅))))
8	7	simplrd 782	. . . 4 ⊢ (𝑅 ∈ DirRel → ( I ↾ ∪ ∪ 𝑅) ⊆ 𝑅)
9	4, 8	eqsstrd 3965	. . 3 ⊢ (𝑅 ∈ DirRel → ( I ↾ 𝑋) ⊆ 𝑅)
10	9	ssbrd 5148	. 2 ⊢ (𝑅 ∈ DirRel → (𝐴( I ↾ 𝑋)𝐴 → 𝐴𝑅𝐴))
11		eqid 2760	. . 3 ⊢ 𝐴 = 𝐴
12		resieq 5983	. . . 4 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝐴( I ↾ 𝑋)𝐴 ↔ 𝐴 = 𝐴))
13	12	anidms 577	. . 3 ⊢ (𝐴 ∈ 𝑋 → (𝐴( I ↾ 𝑋)𝐴 ↔ 𝐴 = 𝐴))
14	11, 13	mpbiri 261	. 2 ⊢ (𝐴 ∈ 𝑋 → 𝐴( I ↾ 𝑋)𝐴)
15	10, 14	impel 515	1 ⊢ ((𝑅 ∈ DirRel ∧ 𝐴 ∈ 𝑋) → 𝐴𝑅𝐴)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ⊆ wss 3899 ∪ cuni 4867 class class class wbr 5103 I cid 5549 × cxp 5653 ^◡ccnv 5654 dom cdm 5655 ↾ cres 5657 ∘ ccom 5659 Rel wrel 5660 DirRelcdir 18682

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-ext 2732 ax-sep 5251 ax-pr 5398

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-sb 2100 df-clab 2739 df-cleq 2752 df-clel 2835 df-ral 3077 df-rex 3087 df-rab 3413 df-v 3452 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4280 df-if 4483 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-br 5104 df-opab 5168 df-id 5550 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-dir 18684

This theorem is used by: tailini 36995

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