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Mirrors > Home > MPE Home > Th. List > dirdm | Structured version Visualization version GIF version |
Description: A direction's domain is equal to its field. (Contributed by Jeff Hankins, 25-Nov-2009.) (Revised by Mario Carneiro, 22-Nov-2013.) |
Ref | Expression |
---|---|
dirdm | ⊢ (𝑅 ∈ DirRel → dom 𝑅 = ∪ ∪ 𝑅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssun1 4130 | . . . 4 ⊢ dom 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅) | |
2 | dmrnssfld 5923 | . . . 4 ⊢ (dom 𝑅 ∪ ran 𝑅) ⊆ ∪ ∪ 𝑅 | |
3 | 1, 2 | sstri 3951 | . . 3 ⊢ dom 𝑅 ⊆ ∪ ∪ 𝑅 |
4 | 3 | a1i 11 | . 2 ⊢ (𝑅 ∈ DirRel → dom 𝑅 ⊆ ∪ ∪ 𝑅) |
5 | dmresi 6003 | . . 3 ⊢ dom ( I ↾ ∪ ∪ 𝑅) = ∪ ∪ 𝑅 | |
6 | eqid 2736 | . . . . . . 7 ⊢ ∪ ∪ 𝑅 = ∪ ∪ 𝑅 | |
7 | 6 | isdir 18441 | . . . . . 6 ⊢ (𝑅 ∈ DirRel → (𝑅 ∈ DirRel ↔ ((Rel 𝑅 ∧ ( I ↾ ∪ ∪ 𝑅) ⊆ 𝑅) ∧ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ (∪ ∪ 𝑅 × ∪ ∪ 𝑅) ⊆ (◡𝑅 ∘ 𝑅))))) |
8 | 7 | ibi 266 | . . . . 5 ⊢ (𝑅 ∈ DirRel → ((Rel 𝑅 ∧ ( I ↾ ∪ ∪ 𝑅) ⊆ 𝑅) ∧ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ (∪ ∪ 𝑅 × ∪ ∪ 𝑅) ⊆ (◡𝑅 ∘ 𝑅)))) |
9 | 8 | simplrd 768 | . . . 4 ⊢ (𝑅 ∈ DirRel → ( I ↾ ∪ ∪ 𝑅) ⊆ 𝑅) |
10 | dmss 5856 | . . . 4 ⊢ (( I ↾ ∪ ∪ 𝑅) ⊆ 𝑅 → dom ( I ↾ ∪ ∪ 𝑅) ⊆ dom 𝑅) | |
11 | 9, 10 | syl 17 | . . 3 ⊢ (𝑅 ∈ DirRel → dom ( I ↾ ∪ ∪ 𝑅) ⊆ dom 𝑅) |
12 | 5, 11 | eqsstrrid 3991 | . 2 ⊢ (𝑅 ∈ DirRel → ∪ ∪ 𝑅 ⊆ dom 𝑅) |
13 | 4, 12 | eqssd 3959 | 1 ⊢ (𝑅 ∈ DirRel → dom 𝑅 = ∪ ∪ 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∪ cun 3906 ⊆ wss 3908 ∪ cuni 4863 I cid 5528 × cxp 5629 ◡ccnv 5630 dom cdm 5631 ran crn 5632 ↾ cres 5633 ∘ ccom 5635 Rel wrel 5636 DirRelcdir 18437 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2707 ax-sep 5254 ax-nul 5261 ax-pr 5382 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-br 5104 df-opab 5166 df-id 5529 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-dir 18439 |
This theorem is referenced by: dirref 18444 dirge 18446 tailfval 34776 tailf 34779 filnetlem4 34785 |
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