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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 17842 | . . 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 2114 ⊆ wss 3936 ∪ cuni 4838 I cid 5459 × cxp 5553 ◡ccnv 5554 ↾ cres 5557 ∘ ccom 5559 Rel wrel 5560 DirRelcdir 17838 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-in 3943 df-ss 3952 df-uni 4839 df-br 5067 df-opab 5129 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-res 5567 df-dir 17840 |
This theorem is referenced by: dirtr 17846 dirge 17847 |
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