Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > reldir | Structured version Visualization version GIF version |
Description: A direction is a relation. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 22-Nov-2013.) |
Ref | Expression |
---|---|
reldir | ⊢ (𝑅 ∈ DirRel → Rel 𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2821 | . . . 4 ⊢ ∪ ∪ 𝑅 = ∪ ∪ 𝑅 | |
2 | 1 | isdir 17836 | . . 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 766 | 1 ⊢ (𝑅 ∈ DirRel → Rel 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2110 ⊆ wss 3936 ∪ cuni 4832 I cid 5454 × cxp 5548 ◡ccnv 5549 ↾ cres 5552 ∘ ccom 5554 Rel wrel 5555 DirRelcdir 17832 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rex 3144 df-rab 3147 df-in 3943 df-ss 3952 df-uni 4833 df-br 5060 df-opab 5122 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-res 5562 df-dir 17834 |
This theorem is referenced by: dirtr 17840 dirge 17841 |
Copyright terms: Public domain | W3C validator |