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Theorem dirref 18524

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 18523	. . . . . 6 ⊢ (𝑅 ∈ DirRel → dom 𝑅 = ∪ ∪ 𝑅)
3	1, 2	eqtrid 2783	. . . . 5 ⊢ (𝑅 ∈ DirRel → 𝑋 = ∪ ∪ 𝑅)
4	3	reseq2d 5938	. . . 4 ⊢ (𝑅 ∈ DirRel → ( I ↾ 𝑋) = ( I ↾ ∪ ∪ 𝑅))
5		eqid 2736	. . . . . . 7 ⊢ ∪ ∪ 𝑅 = ∪ ∪ 𝑅
6	5	isdir 18521	. . . . . 6 ⊢ (𝑅 ∈ DirRel → (𝑅 ∈ DirRel ↔ ((Rel 𝑅 ∧ ( I ↾ ∪ ∪ 𝑅) ⊆ 𝑅) ∧ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ (∪ ∪ 𝑅 × ∪ ∪ 𝑅) ⊆ (^◡𝑅 ∘ 𝑅)))))
7	6	ibi 267	. . . . 5 ⊢ (𝑅 ∈ DirRel → ((Rel 𝑅 ∧ ( I ↾ ∪ ∪ 𝑅) ⊆ 𝑅) ∧ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ (∪ ∪ 𝑅 × ∪ ∪ 𝑅) ⊆ (^◡𝑅 ∘ 𝑅))))
8	7	simplrd 769	. . . 4 ⊢ (𝑅 ∈ DirRel → ( I ↾ ∪ ∪ 𝑅) ⊆ 𝑅)
9	4, 8	eqsstrd 3968	. . 3 ⊢ (𝑅 ∈ DirRel → ( I ↾ 𝑋) ⊆ 𝑅)
10	9	ssbrd 5141	. 2 ⊢ (𝑅 ∈ DirRel → (𝐴( I ↾ 𝑋)𝐴 → 𝐴𝑅𝐴))
11		eqid 2736	. . 3 ⊢ 𝐴 = 𝐴
12		resieq 5949	. . . 4 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝐴( I ↾ 𝑋)𝐴 ↔ 𝐴 = 𝐴))
13	12	anidms 566	. . 3 ⊢ (𝐴 ∈ 𝑋 → (𝐴( I ↾ 𝑋)𝐴 ↔ 𝐴 = 𝐴))
14	11, 13	mpbiri 258	. 2 ⊢ (𝐴 ∈ 𝑋 → 𝐴( I ↾ 𝑋)𝐴)
15	10, 14	impel 505	1 ⊢ ((𝑅 ∈ DirRel ∧ 𝐴 ∈ 𝑋) → 𝐴𝑅𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1541 ∈ wcel 2113 ⊆ wss 3901 ∪ cuni 4863 class class class wbr 5098 I cid 5518 × cxp 5622 ^◡ccnv 5623 dom cdm 5624 ↾ cres 5626 ∘ ccom 5628 Rel wrel 5629 DirRelcdir 18517

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-ext 2708 ax-sep 5241 ax-nul 5251 ax-pr 5377

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-sb 2068 df-clab 2715 df-cleq 2728 df-clel 2811 df-ral 3052 df-rex 3061 df-rab 3400 df-v 3442 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-nul 4286 df-if 4480 df-sn 4581 df-pr 4583 df-op 4587 df-uni 4864 df-br 5099 df-opab 5161 df-id 5519 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-dir 18519

This theorem is referenced by: tailini 36570

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