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Theorem mopnval 23367
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.)
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 6767 . . 3 (∞Met‘𝑋) ⊆ ran ∞Met
21sseli 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))
54rneqd 5824 . . . . 5 (𝑑 = 𝐷 → ran (ball‘𝑑) = ran (ball‘𝐷))
65fveq2d 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
96, 7, 8fvmpt 6839 . . 3 (𝐷 ran ∞Met → (MetOpen‘𝐷) = (topGen‘ran (ball‘𝐷)))
103, 9syl5eq 2792 . 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 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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