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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 4178 | . . . 4 ⊢ dom 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅) | |
| 2 | dmrnssfld 5984 | . . . 4 ⊢ (dom 𝑅 ∪ ran 𝑅) ⊆ ∪ ∪ 𝑅 | |
| 3 | 1, 2 | sstri 3993 | . . 3 ⊢ dom 𝑅 ⊆ ∪ ∪ 𝑅 |
| 4 | 3 | a1i 11 | . 2 ⊢ (𝑅 ∈ DirRel → dom 𝑅 ⊆ ∪ ∪ 𝑅) |
| 5 | dmresi 6070 | . . 3 ⊢ dom ( I ↾ ∪ ∪ 𝑅) = ∪ ∪ 𝑅 | |
| 6 | eqid 2737 | . . . . . . 7 ⊢ ∪ ∪ 𝑅 = ∪ ∪ 𝑅 | |
| 7 | 6 | isdir 18643 | . . . . . 6 ⊢ (𝑅 ∈ DirRel → (𝑅 ∈ DirRel ↔ ((Rel 𝑅 ∧ ( I ↾ ∪ ∪ 𝑅) ⊆ 𝑅) ∧ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ (∪ ∪ 𝑅 × ∪ ∪ 𝑅) ⊆ (◡𝑅 ∘ 𝑅))))) |
| 8 | 7 | ibi 267 | . . . . 5 ⊢ (𝑅 ∈ DirRel → ((Rel 𝑅 ∧ ( I ↾ ∪ ∪ 𝑅) ⊆ 𝑅) ∧ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ (∪ ∪ 𝑅 × ∪ ∪ 𝑅) ⊆ (◡𝑅 ∘ 𝑅)))) |
| 9 | 8 | simplrd 770 | . . . 4 ⊢ (𝑅 ∈ DirRel → ( I ↾ ∪ ∪ 𝑅) ⊆ 𝑅) |
| 10 | dmss 5913 | . . . 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 4023 | . 2 ⊢ (𝑅 ∈ DirRel → ∪ ∪ 𝑅 ⊆ dom 𝑅) |
| 13 | 4, 12 | eqssd 4001 | 1 ⊢ (𝑅 ∈ DirRel → dom 𝑅 = ∪ ∪ 𝑅) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∪ cun 3949 ⊆ wss 3951 ∪ cuni 4907 I cid 5577 × cxp 5683 ◡ccnv 5684 dom cdm 5685 ran crn 5686 ↾ cres 5687 ∘ ccom 5689 Rel wrel 5690 DirRelcdir 18639 |
| 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 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-dir 18641 |
| This theorem is referenced by: dirref 18646 dirge 18648 tailfval 36373 tailf 36376 filnetlem4 36382 |
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