MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mopnval Structured version   Visualization version   GIF version

Theorem mopnval 24463
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 24465, the open sets of a metric space form a topology 𝐽, whose base set is 𝐽 by mopnuni 24466. (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 6939 . . 3 (∞Met‘𝑋) ⊆ ran ∞Met
21sseli 3990 . 2 (𝐷 ∈ (∞Met‘𝑋) → 𝐷 ran ∞Met)
3 mopnval.1 . . 3 𝐽 = (MetOpen‘𝐷)
4 fveq2 6906 . . . . . 6 (𝑑 = 𝐷 → (ball‘𝑑) = (ball‘𝐷))
54rneqd 5951 . . . . 5 (𝑑 = 𝐷 → ran (ball‘𝑑) = ran (ball‘𝐷))
65fveq2d 6910 . . . 4 (𝑑 = 𝐷 → (topGen‘ran (ball‘𝑑)) = (topGen‘ran (ball‘𝐷)))
7 df-mopn 21377 . . . 4 MetOpen = (𝑑 ran ∞Met ↦ (topGen‘ran (ball‘𝑑)))
8 fvex 6919 . . . 4 (topGen‘ran (ball‘𝐷)) ∈ V
96, 7, 8fvmpt 7015 . . 3 (𝐷 ran ∞Met → (MetOpen‘𝐷) = (topGen‘ran (ball‘𝐷)))
103, 9eqtrid 2786 . 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 1536  wcel 2105   cuni 4911  ran crn 5689  cfv 6562  topGenctg 17483  ∞Metcxmet 21366  ballcbl 21368  MetOpencmopn 21371
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-iota 6515  df-fun 6564  df-fv 6570  df-mopn 21377
This theorem is referenced by:  mopntopon  24464  elmopn  24467  imasf1oxms  24517  blssopn  24523  metss  24536  prdsxmslem2  24557  metcnp3  24568  xmetutop  24596  tgioo  24831  ismtyhmeolem  37790
  Copyright terms: Public domain W3C validator