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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 23369, the open sets of a metric space form a topology 𝐽, whose base set is ∪ 𝐽 by mopnuni 23370. (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 6767 | . . 3 ⊢ (∞Met‘𝑋) ⊆ ∪ ran ∞Met | |
2 | 1 | sseli 3913 | . 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷 ∈ ∪ ran ∞Met) |
3 | mopnval.1 | . . 3 ⊢ 𝐽 = (MetOpen‘𝐷) | |
4 | fveq2 6738 | . . . . . 6 ⊢ (𝑑 = 𝐷 → (ball‘𝑑) = (ball‘𝐷)) | |
5 | 4 | rneqd 5824 | . . . . 5 ⊢ (𝑑 = 𝐷 → ran (ball‘𝑑) = ran (ball‘𝐷)) |
6 | 5 | fveq2d 6742 | . . . 4 ⊢ (𝑑 = 𝐷 → (topGen‘ran (ball‘𝑑)) = (topGen‘ran (ball‘𝐷))) |
7 | df-mopn 20391 | . . . 4 ⊢ MetOpen = (𝑑 ∈ ∪ ran ∞Met ↦ (topGen‘ran (ball‘𝑑))) | |
8 | fvex 6751 | . . . 4 ⊢ (topGen‘ran (ball‘𝐷)) ∈ V | |
9 | 6, 7, 8 | fvmpt 6839 | . . 3 ⊢ (𝐷 ∈ ∪ ran ∞Met → (MetOpen‘𝐷) = (topGen‘ran (ball‘𝐷))) |
10 | 3, 9 | syl5eq 2792 | . 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 1543 ∈ wcel 2112 ∪ cuni 4835 ran crn 5569 ‘cfv 6400 topGenctg 16974 ∞Metcxmet 20380 ballcbl 20382 MetOpencmopn 20385 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2710 ax-sep 5208 ax-nul 5215 ax-pr 5338 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2818 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3425 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4254 df-if 4456 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4836 df-br 5070 df-opab 5132 df-mpt 5152 df-id 5471 df-xp 5574 df-rel 5575 df-cnv 5576 df-co 5577 df-dm 5578 df-rn 5579 df-iota 6358 df-fun 6402 df-fv 6408 df-mopn 20391 |
This theorem is referenced by: mopntopon 23368 elmopn 23371 imasf1oxms 23418 blssopn 23424 metss 23437 prdsxmslem2 23458 metcnp3 23469 xmetutop 23497 tgioo 23724 ismtyhmeolem 35735 |
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