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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 24450, the open sets of a metric space form a topology 𝐽, whose base set is ∪ 𝐽 by mopnuni 24451. (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 6939 | . . 3 ⊢ (∞Met‘𝑋) ⊆ ∪ ran ∞Met | |
| 2 | 1 | sseli 3979 | . 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷 ∈ ∪ ran ∞Met) |
| 3 | mopnval.1 | . . 3 ⊢ 𝐽 = (MetOpen‘𝐷) | |
| 4 | fveq2 6906 | . . . . . 6 ⊢ (𝑑 = 𝐷 → (ball‘𝑑) = (ball‘𝐷)) | |
| 5 | 4 | rneqd 5949 | . . . . 5 ⊢ (𝑑 = 𝐷 → ran (ball‘𝑑) = ran (ball‘𝐷)) |
| 6 | 5 | fveq2d 6910 | . . . 4 ⊢ (𝑑 = 𝐷 → (topGen‘ran (ball‘𝑑)) = (topGen‘ran (ball‘𝐷))) |
| 7 | df-mopn 21360 | . . . 4 ⊢ MetOpen = (𝑑 ∈ ∪ ran ∞Met ↦ (topGen‘ran (ball‘𝑑))) | |
| 8 | fvex 6919 | . . . 4 ⊢ (topGen‘ran (ball‘𝐷)) ∈ V | |
| 9 | 6, 7, 8 | fvmpt 7016 | . . 3 ⊢ (𝐷 ∈ ∪ ran ∞Met → (MetOpen‘𝐷) = (topGen‘ran (ball‘𝐷))) |
| 10 | 3, 9 | eqtrid 2789 | . 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 2108 ∪ cuni 4907 ran crn 5686 ‘cfv 6561 topGenctg 17482 ∞Metcxmet 21349 ballcbl 21351 MetOpencmopn 21354 |
| 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 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-iota 6514 df-fun 6563 df-fv 6569 df-mopn 21360 |
| This theorem is referenced by: mopntopon 24449 elmopn 24452 imasf1oxms 24502 blssopn 24508 metss 24521 prdsxmslem2 24542 metcnp3 24553 xmetutop 24581 tgioo 24817 ismtyhmeolem 37811 |
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