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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 | eqid 2610 | . . . 4 ⊢ 𝐴 = 𝐴 | |
2 | resieq 5327 | . . . . 5 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝐴( I ↾ 𝑋)𝐴 ↔ 𝐴 = 𝐴)) | |
3 | 2 | anidms 675 | . . . 4 ⊢ (𝐴 ∈ 𝑋 → (𝐴( I ↾ 𝑋)𝐴 ↔ 𝐴 = 𝐴)) |
4 | 1, 3 | mpbiri 247 | . . 3 ⊢ (𝐴 ∈ 𝑋 → 𝐴( I ↾ 𝑋)𝐴) |
5 | dirref.1 | . . . . . . 7 ⊢ 𝑋 = dom 𝑅 | |
6 | dirdm 17057 | . . . . . . 7 ⊢ (𝑅 ∈ DirRel → dom 𝑅 = ∪ ∪ 𝑅) | |
7 | 5, 6 | syl5eq 2656 | . . . . . 6 ⊢ (𝑅 ∈ DirRel → 𝑋 = ∪ ∪ 𝑅) |
8 | 7 | reseq2d 5317 | . . . . 5 ⊢ (𝑅 ∈ DirRel → ( I ↾ 𝑋) = ( I ↾ ∪ ∪ 𝑅)) |
9 | eqid 2610 | . . . . . . . . 9 ⊢ ∪ ∪ 𝑅 = ∪ ∪ 𝑅 | |
10 | 9 | isdir 17055 | . . . . . . . 8 ⊢ (𝑅 ∈ DirRel → (𝑅 ∈ DirRel ↔ ((Rel 𝑅 ∧ ( I ↾ ∪ ∪ 𝑅) ⊆ 𝑅) ∧ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ (∪ ∪ 𝑅 × ∪ ∪ 𝑅) ⊆ (◡𝑅 ∘ 𝑅))))) |
11 | 10 | ibi 255 | . . . . . . 7 ⊢ (𝑅 ∈ DirRel → ((Rel 𝑅 ∧ ( I ↾ ∪ ∪ 𝑅) ⊆ 𝑅) ∧ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ (∪ ∪ 𝑅 × ∪ ∪ 𝑅) ⊆ (◡𝑅 ∘ 𝑅)))) |
12 | 11 | simpld 474 | . . . . . 6 ⊢ (𝑅 ∈ DirRel → (Rel 𝑅 ∧ ( I ↾ ∪ ∪ 𝑅) ⊆ 𝑅)) |
13 | 12 | simprd 478 | . . . . 5 ⊢ (𝑅 ∈ DirRel → ( I ↾ ∪ ∪ 𝑅) ⊆ 𝑅) |
14 | 8, 13 | eqsstrd 3602 | . . . 4 ⊢ (𝑅 ∈ DirRel → ( I ↾ 𝑋) ⊆ 𝑅) |
15 | 14 | ssbrd 4626 | . . 3 ⊢ (𝑅 ∈ DirRel → (𝐴( I ↾ 𝑋)𝐴 → 𝐴𝑅𝐴)) |
16 | 4, 15 | syl5 33 | . 2 ⊢ (𝑅 ∈ DirRel → (𝐴 ∈ 𝑋 → 𝐴𝑅𝐴)) |
17 | 16 | imp 444 | 1 ⊢ ((𝑅 ∈ DirRel ∧ 𝐴 ∈ 𝑋) → 𝐴𝑅𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ⊆ wss 3540 ∪ cuni 4372 class class class wbr 4583 I cid 4948 × cxp 5036 ◡ccnv 5037 dom cdm 5038 ↾ 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-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-dir 17053 |
This theorem is referenced by: tailini 31541 |
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