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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 24166, the open sets of a metric space form a topology 𝐽, whose base set is ∪ 𝐽 by mopnuni 24167. (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 6923 | . . 3 ⊢ (∞Met‘𝑋) ⊆ ∪ ran ∞Met | |
2 | 1 | sseli 3977 | . 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷 ∈ ∪ ran ∞Met) |
3 | mopnval.1 | . . 3 ⊢ 𝐽 = (MetOpen‘𝐷) | |
4 | fveq2 6890 | . . . . . 6 ⊢ (𝑑 = 𝐷 → (ball‘𝑑) = (ball‘𝐷)) | |
5 | 4 | rneqd 5936 | . . . . 5 ⊢ (𝑑 = 𝐷 → ran (ball‘𝑑) = ran (ball‘𝐷)) |
6 | 5 | fveq2d 6894 | . . . 4 ⊢ (𝑑 = 𝐷 → (topGen‘ran (ball‘𝑑)) = (topGen‘ran (ball‘𝐷))) |
7 | df-mopn 21140 | . . . 4 ⊢ MetOpen = (𝑑 ∈ ∪ ran ∞Met ↦ (topGen‘ran (ball‘𝑑))) | |
8 | fvex 6903 | . . . 4 ⊢ (topGen‘ran (ball‘𝐷)) ∈ V | |
9 | 6, 7, 8 | fvmpt 6997 | . . 3 ⊢ (𝐷 ∈ ∪ ran ∞Met → (MetOpen‘𝐷) = (topGen‘ran (ball‘𝐷))) |
10 | 3, 9 | eqtrid 2782 | . 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 1539 ∈ wcel 2104 ∪ cuni 4907 ran crn 5676 ‘cfv 6542 topGenctg 17387 ∞Metcxmet 21129 ballcbl 21131 MetOpencmopn 21134 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3431 df-v 3474 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-iota 6494 df-fun 6544 df-fv 6550 df-mopn 21140 |
This theorem is referenced by: mopntopon 24165 elmopn 24168 imasf1oxms 24218 blssopn 24224 metss 24237 prdsxmslem2 24258 metcnp3 24269 xmetutop 24297 tgioo 24532 ismtyhmeolem 36975 |
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