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Mirrors > Home > MPE Home > Th. List > dirref | Structured version Visualization version GIF version |
Description: A direction is reflexive. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 22-Nov-2013.) |
Ref | Expression |
---|---|
dirref.1 | ⊢ 𝑋 = dom 𝑅 |
Ref | Expression |
---|---|
dirref | ⊢ ((𝑅 ∈ DirRel ∧ 𝐴 ∈ 𝑋) → 𝐴𝑅𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dirref.1 | . . . . . 6 ⊢ 𝑋 = dom 𝑅 | |
2 | dirdm 18359 | . . . . . 6 ⊢ (𝑅 ∈ DirRel → dom 𝑅 = ∪ ∪ 𝑅) | |
3 | 1, 2 | eqtrid 2788 | . . . . 5 ⊢ (𝑅 ∈ DirRel → 𝑋 = ∪ ∪ 𝑅) |
4 | 3 | reseq2d 5899 | . . . 4 ⊢ (𝑅 ∈ DirRel → ( I ↾ 𝑋) = ( I ↾ ∪ ∪ 𝑅)) |
5 | eqid 2736 | . . . . . . 7 ⊢ ∪ ∪ 𝑅 = ∪ ∪ 𝑅 | |
6 | 5 | isdir 18357 | . . . . . 6 ⊢ (𝑅 ∈ DirRel → (𝑅 ∈ DirRel ↔ ((Rel 𝑅 ∧ ( I ↾ ∪ ∪ 𝑅) ⊆ 𝑅) ∧ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ (∪ ∪ 𝑅 × ∪ ∪ 𝑅) ⊆ (◡𝑅 ∘ 𝑅))))) |
7 | 6 | ibi 268 | . . . . 5 ⊢ (𝑅 ∈ DirRel → ((Rel 𝑅 ∧ ( I ↾ ∪ ∪ 𝑅) ⊆ 𝑅) ∧ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ (∪ ∪ 𝑅 × ∪ ∪ 𝑅) ⊆ (◡𝑅 ∘ 𝑅)))) |
8 | 7 | simplrd 768 | . . . 4 ⊢ (𝑅 ∈ DirRel → ( I ↾ ∪ ∪ 𝑅) ⊆ 𝑅) |
9 | 4, 8 | eqsstrd 3964 | . . 3 ⊢ (𝑅 ∈ DirRel → ( I ↾ 𝑋) ⊆ 𝑅) |
10 | 9 | ssbrd 5124 | . 2 ⊢ (𝑅 ∈ DirRel → (𝐴( I ↾ 𝑋)𝐴 → 𝐴𝑅𝐴)) |
11 | eqid 2736 | . . 3 ⊢ 𝐴 = 𝐴 | |
12 | resieq 5910 | . . . 4 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝐴( I ↾ 𝑋)𝐴 ↔ 𝐴 = 𝐴)) | |
13 | 12 | anidms 568 | . . 3 ⊢ (𝐴 ∈ 𝑋 → (𝐴( I ↾ 𝑋)𝐴 ↔ 𝐴 = 𝐴)) |
14 | 11, 13 | mpbiri 259 | . 2 ⊢ (𝐴 ∈ 𝑋 → 𝐴( I ↾ 𝑋)𝐴) |
15 | 10, 14 | impel 507 | 1 ⊢ ((𝑅 ∈ DirRel ∧ 𝐴 ∈ 𝑋) → 𝐴𝑅𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1539 ∈ wcel 2104 ⊆ wss 3892 ∪ cuni 4844 class class class wbr 5081 I cid 5495 × cxp 5594 ◡ccnv 5595 dom cdm 5596 ↾ cres 5598 ∘ ccom 5600 Rel wrel 5601 DirRelcdir 18353 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-ext 2707 ax-sep 5232 ax-nul 5239 ax-pr 5361 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1780 df-sb 2066 df-clab 2714 df-cleq 2728 df-clel 2814 df-ral 3063 df-rex 3072 df-rab 3287 df-v 3439 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-sn 4566 df-pr 4568 df-op 4572 df-uni 4845 df-br 5082 df-opab 5144 df-id 5496 df-xp 5602 df-rel 5603 df-cnv 5604 df-co 5605 df-dm 5606 df-rn 5607 df-res 5608 df-dir 18355 |
This theorem is referenced by: tailini 34606 |
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