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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 24167, the open sets of a metric space form a topology 𝐽, whose base set is ∪ 𝐽 by mopnuni 24168. (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 6924 | . . 3 ⊢ (∞Met‘𝑋) ⊆ ∪ ran ∞Met | |
2 | 1 | sseli 3978 | . 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷 ∈ ∪ ran ∞Met) |
3 | mopnval.1 | . . 3 ⊢ 𝐽 = (MetOpen‘𝐷) | |
4 | fveq2 6891 | . . . . . 6 ⊢ (𝑑 = 𝐷 → (ball‘𝑑) = (ball‘𝐷)) | |
5 | 4 | rneqd 5937 | . . . . 5 ⊢ (𝑑 = 𝐷 → ran (ball‘𝑑) = ran (ball‘𝐷)) |
6 | 5 | fveq2d 6895 | . . . 4 ⊢ (𝑑 = 𝐷 → (topGen‘ran (ball‘𝑑)) = (topGen‘ran (ball‘𝐷))) |
7 | df-mopn 21141 | . . . 4 ⊢ MetOpen = (𝑑 ∈ ∪ ran ∞Met ↦ (topGen‘ran (ball‘𝑑))) | |
8 | fvex 6904 | . . . 4 ⊢ (topGen‘ran (ball‘𝐷)) ∈ V | |
9 | 6, 7, 8 | fvmpt 6998 | . . 3 ⊢ (𝐷 ∈ ∪ ran ∞Met → (MetOpen‘𝐷) = (topGen‘ran (ball‘𝐷))) |
10 | 3, 9 | eqtrid 2783 | . 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 1540 ∈ wcel 2105 ∪ cuni 4908 ran crn 5677 ‘cfv 6543 topGenctg 17388 ∞Metcxmet 21130 ballcbl 21132 MetOpencmopn 21135 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-iota 6495 df-fun 6545 df-fv 6551 df-mopn 21141 |
This theorem is referenced by: mopntopon 24166 elmopn 24169 imasf1oxms 24219 blssopn 24225 metss 24238 prdsxmslem2 24259 metcnp3 24270 xmetutop 24298 tgioo 24533 ismtyhmeolem 36976 |
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