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Theorem mopnval 22651
 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 22653, the open sets of a metric space form a topology 𝐽, whose base set is ∪ 𝐽 by mopnuni 22654. (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 6475 . . 3 (∞Met‘𝑋) ⊆ ran ∞Met
21sseli 3817 . 2 (𝐷 ∈ (∞Met‘𝑋) → 𝐷 ran ∞Met)
3 mopnval.1 . . 3 𝐽 = (MetOpen‘𝐷)
4 fveq2 6446 . . . . . 6 (𝑑 = 𝐷 → (ball‘𝑑) = (ball‘𝐷))
54rneqd 5598 . . . . 5 (𝑑 = 𝐷 → ran (ball‘𝑑) = ran (ball‘𝐷))
65fveq2d 6450 . . . 4 (𝑑 = 𝐷 → (topGen‘ran (ball‘𝑑)) = (topGen‘ran (ball‘𝐷)))
7 df-mopn 20138 . . . 4 MetOpen = (𝑑 ran ∞Met ↦ (topGen‘ran (ball‘𝑑)))
8 fvex 6459 . . . 4 (topGen‘ran (ball‘𝐷)) ∈ V
96, 7, 8fvmpt 6542 . . 3 (𝐷 ran ∞Met → (MetOpen‘𝐷) = (topGen‘ran (ball‘𝐷)))
103, 9syl5eq 2826 . 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 1601   ∈ wcel 2107  ∪ cuni 4671  ran crn 5356  ‘cfv 6135  topGenctg 16484  ∞Metcxmet 20127  ballcbl 20129  MetOpencmopn 20132 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-iota 6099  df-fun 6137  df-fv 6143  df-mopn 20138 This theorem is referenced by:  mopntopon  22652  elmopn  22655  imasf1oxms  22702  blssopn  22708  metss  22721  prdsxmslem2  22742  metcnp3  22753  xmetutop  22781  tgioo  23007  ismtyhmeolem  34229
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