Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 4150 | . . . 4 ⊢ dom 𝑅 ⊆ (dom 𝑅 ∪ ran 𝑅) | |
2 | dmrnssfld 5843 | . . . 4 ⊢ (dom 𝑅 ∪ ran 𝑅) ⊆ ∪ ∪ 𝑅 | |
3 | 1, 2 | sstri 3978 | . . 3 ⊢ dom 𝑅 ⊆ ∪ ∪ 𝑅 |
4 | 3 | a1i 11 | . 2 ⊢ (𝑅 ∈ DirRel → dom 𝑅 ⊆ ∪ ∪ 𝑅) |
5 | dmresi 5923 | . . 3 ⊢ dom ( I ↾ ∪ ∪ 𝑅) = ∪ ∪ 𝑅 | |
6 | eqid 2823 | . . . . . . 7 ⊢ ∪ ∪ 𝑅 = ∪ ∪ 𝑅 | |
7 | 6 | isdir 17844 | . . . . . 6 ⊢ (𝑅 ∈ DirRel → (𝑅 ∈ DirRel ↔ ((Rel 𝑅 ∧ ( I ↾ ∪ ∪ 𝑅) ⊆ 𝑅) ∧ ((𝑅 ∘ 𝑅) ⊆ 𝑅 ∧ (∪ ∪ 𝑅 × ∪ ∪ 𝑅) ⊆ (◡𝑅 ∘ 𝑅))))) |
8 | 7 | ibi 269 | . . . . 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 5773 | . . . 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 4018 | . 2 ⊢ (𝑅 ∈ DirRel → ∪ ∪ 𝑅 ⊆ dom 𝑅) |
13 | 4, 12 | eqssd 3986 | 1 ⊢ (𝑅 ∈ DirRel → dom 𝑅 = ∪ ∪ 𝑅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∪ cun 3936 ⊆ wss 3938 ∪ cuni 4840 I cid 5461 × cxp 5555 ◡ccnv 5556 dom cdm 5557 ran crn 5558 ↾ cres 5559 ∘ ccom 5561 Rel wrel 5562 DirRelcdir 17840 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-dir 17842 |
This theorem is referenced by: dirref 17847 dirge 17849 tailfval 33722 tailf 33725 filnetlem4 33731 |
Copyright terms: Public domain | W3C validator |