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Theorem mopnval 24388
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 24390, the open sets of a metric space form a topology 𝐽, whose base set is 𝐽 by mopnuni 24391. (Contributed by NM, 1-Sep-2006.) (Revised by Mario Carneiro, 12-Nov-2013.)
Hypothesis
Ref Expression
mopnval.1 𝐽 = (MetOpen‘𝐷)
Assertion
Ref Expression
mopnval (𝐷 ∈ (∞Met‘𝑋) → 𝐽 = (topGen‘ran (ball‘𝐷)))

Proof of Theorem mopnval
Dummy variable 𝑑 is distinct from all other variables.
StepHypRef Expression
1 fvssunirn 6929 . . 3 (∞Met‘𝑋) ⊆ ran ∞Met
21sseli 3972 . 2 (𝐷 ∈ (∞Met‘𝑋) → 𝐷 ran ∞Met)
3 mopnval.1 . . 3 𝐽 = (MetOpen‘𝐷)
4 fveq2 6896 . . . . . 6 (𝑑 = 𝐷 → (ball‘𝑑) = (ball‘𝐷))
54rneqd 5940 . . . . 5 (𝑑 = 𝐷 → ran (ball‘𝑑) = ran (ball‘𝐷))
65fveq2d 6900 . . . 4 (𝑑 = 𝐷 → (topGen‘ran (ball‘𝑑)) = (topGen‘ran (ball‘𝐷)))
7 df-mopn 21292 . . . 4 MetOpen = (𝑑 ran ∞Met ↦ (topGen‘ran (ball‘𝑑)))
8 fvex 6909 . . . 4 (topGen‘ran (ball‘𝐷)) ∈ V
96, 7, 8fvmpt 7004 . . 3 (𝐷 ran ∞Met → (MetOpen‘𝐷) = (topGen‘ran (ball‘𝐷)))
103, 9eqtrid 2777 . 2 (𝐷 ran ∞Met → 𝐽 = (topGen‘ran (ball‘𝐷)))
112, 10syl 17 1 (𝐷 ∈ (∞Met‘𝑋) → 𝐽 = (topGen‘ran (ball‘𝐷)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098   cuni 4909  ran crn 5679  cfv 6549  topGenctg 17422  ∞Metcxmet 21281  ballcbl 21283  MetOpencmopn 21286
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-iota 6501  df-fun 6551  df-fv 6557  df-mopn 21292
This theorem is referenced by:  mopntopon  24389  elmopn  24392  imasf1oxms  24442  blssopn  24448  metss  24461  prdsxmslem2  24482  metcnp3  24493  xmetutop  24521  tgioo  24756  ismtyhmeolem  37405
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