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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 2610 | . . . 4 ⊢ ∪ ∪ 𝑅 = ∪ ∪ 𝑅 | |
2 | 1 | isdir 17055 | . . 3 ⊢ (𝑅 ∈ DirRel → (𝑅 ∈ DirRel ↔ ((Rel 𝑅 ∧ ( I ↾ ∪ ∪ 𝑅) ⊆ 𝑅) ∧ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ (∪ ∪ 𝑅 × ∪ ∪ 𝑅) ⊆ (◡𝑅 ∘ 𝑅))))) |
3 | 2 | ibi 255 | . 2 ⊢ (𝑅 ∈ DirRel → ((Rel 𝑅 ∧ ( I ↾ ∪ ∪ 𝑅) ⊆ 𝑅) ∧ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ (∪ ∪ 𝑅 × ∪ ∪ 𝑅) ⊆ (◡𝑅 ∘ 𝑅)))) |
4 | 3 | simplld 787 | 1 ⊢ (𝑅 ∈ DirRel → Rel 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∈ wcel 1977 ⊆ wss 3540 ∪ cuni 4372 I cid 4948 × cxp 5036 ◡ccnv 5037 ↾ cres 5040 ∘ ccom 5042 Rel wrel 5043 DirRelcdir 17051 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rex 2902 df-v 3175 df-in 3547 df-ss 3554 df-uni 4373 df-br 4584 df-opab 4644 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-res 5050 df-dir 17053 |
This theorem is referenced by: dirtr 17059 dirge 17060 |
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