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Theorem dirref 18558

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 18557	. . . . . 6 ⊢ (𝑅 ∈ DirRel → dom 𝑅 = ∪ ∪ 𝑅)
3	1, 2	eqtrid 2784	. . . . 5 ⊢ (𝑅 ∈ DirRel → 𝑋 = ∪ ∪ 𝑅)
4	3	reseq2d 5981	. . . 4 ⊢ (𝑅 ∈ DirRel → ( I ↾ 𝑋) = ( I ↾ ∪ ∪ 𝑅))
5		eqid 2732	. . . . . . 7 ⊢ ∪ ∪ 𝑅 = ∪ ∪ 𝑅
6	5	isdir 18555	. . . . . 6 ⊢ (𝑅 ∈ DirRel → (𝑅 ∈ DirRel ↔ ((Rel 𝑅 ∧ ( I ↾ ∪ ∪ 𝑅) ⊆ 𝑅) ∧ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ (∪ ∪ 𝑅 × ∪ ∪ 𝑅) ⊆ (^◡𝑅 ∘ 𝑅)))))
7	6	ibi 266	. . . . 5 ⊢ (𝑅 ∈ DirRel → ((Rel 𝑅 ∧ ( I ↾ ∪ ∪ 𝑅) ⊆ 𝑅) ∧ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ (∪ ∪ 𝑅 × ∪ ∪ 𝑅) ⊆ (^◡𝑅 ∘ 𝑅))))
8	7	simplrd 768	. . . 4 ⊢ (𝑅 ∈ DirRel → ( I ↾ ∪ ∪ 𝑅) ⊆ 𝑅)
9	4, 8	eqsstrd 4020	. . 3 ⊢ (𝑅 ∈ DirRel → ( I ↾ 𝑋) ⊆ 𝑅)
10	9	ssbrd 5191	. 2 ⊢ (𝑅 ∈ DirRel → (𝐴( I ↾ 𝑋)𝐴 → 𝐴𝑅𝐴))
11		eqid 2732	. . 3 ⊢ 𝐴 = 𝐴
12		resieq 5992	. . . 4 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝐴( I ↾ 𝑋)𝐴 ↔ 𝐴 = 𝐴))
13	12	anidms 567	. . 3 ⊢ (𝐴 ∈ 𝑋 → (𝐴( I ↾ 𝑋)𝐴 ↔ 𝐴 = 𝐴))
14	11, 13	mpbiri 257	. 2 ⊢ (𝐴 ∈ 𝑋 → 𝐴( I ↾ 𝑋)𝐴)
15	10, 14	impel 506	1 ⊢ ((𝑅 ∈ DirRel ∧ 𝐴 ∈ 𝑋) → 𝐴𝑅𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ⊆ wss 3948 ∪ cuni 4908 class class class wbr 5148 I cid 5573 × cxp 5674 ^◡ccnv 5675 dom cdm 5676 ↾ cres 5678 ∘ ccom 5680 Rel wrel 5681 DirRelcdir 18551

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-dir 18553

This theorem is referenced by: tailini 35564

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