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Mirrors > Home > MPE Home > Th. List > mopnval | Structured version Visualization version GIF version |
Description: An open set is a subset of a metric space which includes a ball around each of its points. Definition 1.3-2 of [Kreyszig] p. 18. The object (MetOpen‘𝐷) is the family of all open sets in the metric space determined by the metric 𝐷. By mopntop 22653, the open sets of a metric space form a topology 𝐽, whose base set is ∪ 𝐽 by mopnuni 22654. (Contributed by NM, 1-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.) |
Ref | Expression |
---|---|
mopnval.1 | ⊢ 𝐽 = (MetOpen‘𝐷) |
Ref | Expression |
---|---|
mopnval | ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 = (topGen‘ran (ball‘𝐷))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvssunirn 6475 | . . 3 ⊢ (∞Met‘𝑋) ⊆ ∪ ran ∞Met | |
2 | 1 | sseli 3817 | . 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷 ∈ ∪ ran ∞Met) |
3 | mopnval.1 | . . 3 ⊢ 𝐽 = (MetOpen‘𝐷) | |
4 | fveq2 6446 | . . . . . 6 ⊢ (𝑑 = 𝐷 → (ball‘𝑑) = (ball‘𝐷)) | |
5 | 4 | rneqd 5598 | . . . . 5 ⊢ (𝑑 = 𝐷 → ran (ball‘𝑑) = ran (ball‘𝐷)) |
6 | 5 | fveq2d 6450 | . . . 4 ⊢ (𝑑 = 𝐷 → (topGen‘ran (ball‘𝑑)) = (topGen‘ran (ball‘𝐷))) |
7 | df-mopn 20138 | . . . 4 ⊢ MetOpen = (𝑑 ∈ ∪ ran ∞Met ↦ (topGen‘ran (ball‘𝑑))) | |
8 | fvex 6459 | . . . 4 ⊢ (topGen‘ran (ball‘𝐷)) ∈ V | |
9 | 6, 7, 8 | fvmpt 6542 | . . 3 ⊢ (𝐷 ∈ ∪ ran ∞Met → (MetOpen‘𝐷) = (topGen‘ran (ball‘𝐷))) |
10 | 3, 9 | syl5eq 2826 | . 2 ⊢ (𝐷 ∈ ∪ ran ∞Met → 𝐽 = (topGen‘ran (ball‘𝐷))) |
11 | 2, 10 | syl 17 | 1 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 = (topGen‘ran (ball‘𝐷))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 ∈ wcel 2107 ∪ cuni 4671 ran crn 5356 ‘cfv 6135 topGenctg 16484 ∞Metcxmet 20127 ballcbl 20129 MetOpencmopn 20132 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-sbc 3653 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4672 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-iota 6099 df-fun 6137 df-fv 6143 df-mopn 20138 |
This theorem is referenced by: mopntopon 22652 elmopn 22655 imasf1oxms 22702 blssopn 22708 metss 22721 prdsxmslem2 22742 metcnp3 22753 xmetutop 22781 tgioo 23007 ismtyhmeolem 34229 |
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