Theorem reldir 18656

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 2763	. . . 4 ⊢ ∪ ∪ 𝑅 = ∪ ∪ 𝑅
2	1	isdir 18655	. . 3 ⊢ (𝑅 ∈ DirRel → (𝑅 ∈ DirRel ↔ ((Rel 𝑅 ∧ ( I ↾ ∪ ∪ 𝑅) ⊆ 𝑅) ∧ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ (∪ ∪ 𝑅 × ∪ ∪ 𝑅) ⊆ (◡𝑅 ∘ 𝑅)))))
3	2	ibi 270	. 2 ⊢ (𝑅 ∈ DirRel → ((Rel 𝑅 ∧ ( I ↾ ∪ ∪ 𝑅) ⊆ 𝑅) ∧ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ (∪ ∪ 𝑅 × ∪ ∪ 𝑅) ⊆ (◡𝑅 ∘ 𝑅))))
4	3	simplld 779	1 ⊢ (𝑅 ∈ DirRel → Rel 𝑅)

Syntax hints:   → wi 4   ∧ wa 400   ∈ wcel 2143   ⊆ wss 3906  ∪ cuni 4873   I cid 5557   × cxp 5661  ^◡ccnv 5662   ↾ cres 5665   ∘ ccom 5667  Rel wrel 5668  DirRelcdir 18651

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3417  df-v 3457  df-in 3913  df-ss 3923  df-uni 4874  df-br 5111  df-opab 5175  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-res 5675  df-dir 18653

This theorem is referenced by:  dirtr  18659  dirge  18660