Theorem reldir 18693

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 2762	. . . 4 ⊢ ∪ ∪ 𝑅 = ∪ ∪ 𝑅
2	1	isdir 18692	. . 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 780	1 ⊢ (𝑅 ∈ DirRel → Rel 𝑅)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∈ wcel 2145   ⊆ wss 3902  ∪ cuni 4870   I cid 5553   × cxp 5657  ^◡ccnv 5658   ↾ cres 5661   ∘ ccom 5663  Rel wrel 5664  DirRelcdir 18688

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3415  df-v 3455  df-in 3909  df-ss 3919  df-uni 4871  df-br 5108  df-opab 5172  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-res 5671  df-dir 18690

This theorem is used by:  dirtr  18696  dirge  18697