Theorem mopnval 24469

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 24471, the open sets of a metric space form a topology 𝐽, whose base set is ∪ 𝐽 by mopnuni 24472. (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.

Step	Hyp	Ref	Expression
1		fvssunirn 6953	. . 3 ⊢ (∞Met‘𝑋) ⊆ ∪ ran ∞Met
2	1	sseli 4004	. 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷 ∈ ∪ ran ∞Met)
3		mopnval.1	. . 3 ⊢ 𝐽 = (MetOpen‘𝐷)
4		fveq2 6920	. . . . . 6 ⊢ (𝑑 = 𝐷 → (ball‘𝑑) = (ball‘𝐷))
5	4	rneqd 5963	. . . . 5 ⊢ (𝑑 = 𝐷 → ran (ball‘𝑑) = ran (ball‘𝐷))
6	5	fveq2d 6924	. . . 4 ⊢ (𝑑 = 𝐷 → (topGen‘ran (ball‘𝑑)) = (topGen‘ran (ball‘𝐷)))
7		df-mopn 21383	. . . 4 ⊢ MetOpen = (𝑑 ∈ ∪ ran ∞Met ↦ (topGen‘ran (ball‘𝑑)))
8		fvex 6933	. . . 4 ⊢ (topGen‘ran (ball‘𝐷)) ∈ V
9	6, 7, 8	fvmpt 7029	. . 3 ⊢ (𝐷 ∈ ∪ ran ∞Met → (MetOpen‘𝐷) = (topGen‘ran (ball‘𝐷)))
10	3, 9	eqtrid 2792	. 2 ⊢ (𝐷 ∈ ∪ ran ∞Met → 𝐽 = (topGen‘ran (ball‘𝐷)))
11	2, 10	syl 17	1 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 = (topGen‘ran (ball‘𝐷)))

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2108  ∪ cuni 4931  ran crn 5701  ‘cfv 6573  topGenctg 17497  ∞Metcxmet 21372  ballcbl 21374  MetOpencmopn 21377

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-iota 6525  df-fun 6575  df-fv 6581  df-mopn 21383

This theorem is referenced by:  mopntopon  24470  elmopn  24473  imasf1oxms  24523  blssopn  24529  metss  24542  prdsxmslem2  24563  metcnp3  24574  xmetutop  24602  tgioo  24837  ismtyhmeolem  37764