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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 18657 | . . . . . 6 ⊢ (𝑅 ∈ DirRel → dom 𝑅 = ∪ ∪ 𝑅) | |
3 | 1, 2 | eqtrid 2786 | . . . . 5 ⊢ (𝑅 ∈ DirRel → 𝑋 = ∪ ∪ 𝑅) |
4 | 3 | reseq2d 5999 | . . . 4 ⊢ (𝑅 ∈ DirRel → ( I ↾ 𝑋) = ( I ↾ ∪ ∪ 𝑅)) |
5 | eqid 2734 | . . . . . . 7 ⊢ ∪ ∪ 𝑅 = ∪ ∪ 𝑅 | |
6 | 5 | isdir 18655 | . . . . . 6 ⊢ (𝑅 ∈ DirRel → (𝑅 ∈ DirRel ↔ ((Rel 𝑅 ∧ ( I ↾ ∪ ∪ 𝑅) ⊆ 𝑅) ∧ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ (∪ ∪ 𝑅 × ∪ ∪ 𝑅) ⊆ (◡𝑅 ∘ 𝑅))))) |
7 | 6 | ibi 267 | . . . . 5 ⊢ (𝑅 ∈ DirRel → ((Rel 𝑅 ∧ ( I ↾ ∪ ∪ 𝑅) ⊆ 𝑅) ∧ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ (∪ ∪ 𝑅 × ∪ ∪ 𝑅) ⊆ (◡𝑅 ∘ 𝑅)))) |
8 | 7 | simplrd 770 | . . . 4 ⊢ (𝑅 ∈ DirRel → ( I ↾ ∪ ∪ 𝑅) ⊆ 𝑅) |
9 | 4, 8 | eqsstrd 4033 | . . 3 ⊢ (𝑅 ∈ DirRel → ( I ↾ 𝑋) ⊆ 𝑅) |
10 | 9 | ssbrd 5190 | . 2 ⊢ (𝑅 ∈ DirRel → (𝐴( I ↾ 𝑋)𝐴 → 𝐴𝑅𝐴)) |
11 | eqid 2734 | . . 3 ⊢ 𝐴 = 𝐴 | |
12 | resieq 6010 | . . . 4 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝐴( I ↾ 𝑋)𝐴 ↔ 𝐴 = 𝐴)) | |
13 | 12 | anidms 566 | . . 3 ⊢ (𝐴 ∈ 𝑋 → (𝐴( I ↾ 𝑋)𝐴 ↔ 𝐴 = 𝐴)) |
14 | 11, 13 | mpbiri 258 | . 2 ⊢ (𝐴 ∈ 𝑋 → 𝐴( I ↾ 𝑋)𝐴) |
15 | 10, 14 | impel 505 | 1 ⊢ ((𝑅 ∈ DirRel ∧ 𝐴 ∈ 𝑋) → 𝐴𝑅𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1536 ∈ wcel 2105 ⊆ wss 3962 ∪ cuni 4911 class class class wbr 5147 I cid 5581 × cxp 5686 ◡ccnv 5687 dom cdm 5688 ↾ cres 5690 ∘ ccom 5692 Rel wrel 5693 DirRelcdir 18651 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-dir 18653 |
This theorem is referenced by: tailini 36358 |
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