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Theorem mopnval 22183
 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 22185, the open sets of a metric space form a topology 𝐽, whose base set is ∪ 𝐽 by mopnuni 22186. (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 6184 . . 3 (∞Met‘𝑋) ⊆ ran ∞Met
21sseli 3584 . 2 (𝐷 ∈ (∞Met‘𝑋) → 𝐷 ran ∞Met)
3 mopnval.1 . . 3 𝐽 = (MetOpen‘𝐷)
4 fveq2 6158 . . . . . 6 (𝑑 = 𝐷 → (ball‘𝑑) = (ball‘𝐷))
54rneqd 5323 . . . . 5 (𝑑 = 𝐷 → ran (ball‘𝑑) = ran (ball‘𝐷))
65fveq2d 6162 . . . 4 (𝑑 = 𝐷 → (topGen‘ran (ball‘𝑑)) = (topGen‘ran (ball‘𝐷)))
7 df-mopn 19682 . . . 4 MetOpen = (𝑑 ran ∞Met ↦ (topGen‘ran (ball‘𝑑)))
8 fvex 6168 . . . 4 (topGen‘ran (ball‘𝐷)) ∈ V
96, 7, 8fvmpt 6249 . . 3 (𝐷 ran ∞Met → (MetOpen‘𝐷) = (topGen‘ran (ball‘𝐷)))
103, 9syl5eq 2667 . 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 1480   ∈ wcel 1987  ∪ cuni 4409  ran crn 5085  ‘cfv 5857  topGenctg 16038  ∞Metcxmt 19671  ballcbl 19673  MetOpencmopn 19676 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-iota 5820  df-fun 5859  df-fv 5865  df-mopn 19682 This theorem is referenced by:  mopntopon  22184  elmopn  22187  imasf1oxms  22234  blssopn  22240  metss  22253  prdsxmslem2  22274  metcnp3  22285  xmetutop  22313  tgioo  22539  ismtyhmeolem  33274
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