Theorem reldir 18557

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 18556	. . 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 765	1 ⊢ (𝑅 ∈ DirRel → Rel 𝑅)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2105   ⊆ wss 3948  ∪ cuni 4908   I cid 5573   × cxp 5674  ^◡ccnv 5675   ↾ cres 5678   ∘ ccom 5680  Rel wrel 5681  DirRelcdir 18552

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-rab 3432  df-v 3475  df-in 3955  df-ss 3965  df-uni 4909  df-br 5149  df-opab 5211  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-res 5688  df-dir 18554

This theorem is referenced by:  dirtr  18560  dirge  18561