Theorem reldir 18607

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 2756	. . . 4 ⊢ ∪ ∪ 𝑅 = ∪ ∪ 𝑅
2	1	isdir 18606	. . 3 ⊢ (𝑅 ∈ DirRel → (𝑅 ∈ DirRel ↔ ((Rel 𝑅 ∧ ( I ↾ ∪ ∪ 𝑅) ⊆ 𝑅) ∧ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ (∪ ∪ 𝑅 × ∪ ∪ 𝑅) ⊆ (◡𝑅 ∘ 𝑅)))))
3	2	ibi 269	. 2 ⊢ (𝑅 ∈ DirRel → ((Rel 𝑅 ∧ ( I ↾ ∪ ∪ 𝑅) ⊆ 𝑅) ∧ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ (∪ ∪ 𝑅 × ∪ ∪ 𝑅) ⊆ (◡𝑅 ∘ 𝑅))))
4	3	simplld 775	1 ⊢ (𝑅 ∈ DirRel → Rel 𝑅)

Syntax hints:   → wi 4   ∧ wa 398   ∈ wcel 2136   ⊆ wss 3899  ∪ cuni 4859   I cid 5534   × cxp 5638  ^◡ccnv 5639   ↾ cres 5642   ∘ ccom 5644  Rel wrel 5645  DirRelcdir 18602

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-ext 2728

This theorem depends on definitions:  df-bi 209  df-an 399  df-tru 1557  df-ex 1794  df-sb 2085  df-clab 2735  df-cleq 2748  df-clel 2831  df-rab 3409  df-v 3450  df-in 3906  df-ss 3916  df-uni 4860  df-br 5095  df-opab 5157  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-res 5652  df-dir 18604

This theorem is referenced by:  dirtr  18610  dirge  18611