Theorem reldir 18502

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

Assertion

Ref	Expression
reldir	⊢ (𝑅 ∈ DirRel → Rel 𝑅)

Proof of Theorem reldir

Step	Hyp	Ref	Expression
1		eqid 2731	. . . 4 ⊢ ∪ ∪ 𝑅 = ∪ ∪ 𝑅
2	1	isdir 18501	. . 3 ⊢ (𝑅 ∈ DirRel → (𝑅 ∈ DirRel ↔ ((Rel 𝑅 ∧ ( I ↾ ∪ ∪ 𝑅) ⊆ 𝑅) ∧ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ (∪ ∪ 𝑅 × ∪ ∪ 𝑅) ⊆ (◡𝑅 ∘ 𝑅)))))
3	2	ibi 267	. 2 ⊢ (𝑅 ∈ DirRel → ((Rel 𝑅 ∧ ( I ↾ ∪ ∪ 𝑅) ⊆ 𝑅) ∧ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ (∪ ∪ 𝑅 × ∪ ∪ 𝑅) ⊆ (◡𝑅 ∘ 𝑅))))
4	3	simplld 767	1 ⊢ (𝑅 ∈ DirRel → Rel 𝑅)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2111   ⊆ wss 3902  ∪ cuni 4859   I cid 5510   × cxp 5614  ^◡ccnv 5615   ↾ cres 5618   ∘ ccom 5620  Rel wrel 5621  DirRelcdir 18497

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 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-rab 3396  df-v 3438  df-in 3909  df-ss 3919  df-uni 4860  df-br 5092  df-opab 5154  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-res 5628  df-dir 18499

This theorem is referenced by:  dirtr  18505  dirge  18506