Theorem mopnval 24388

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 24390, the open sets of a metric space form a topology 𝐽, whose base set is ∪ 𝐽 by mopnuni 24391. (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 6929	. . 3 ⊢ (∞Met‘𝑋) ⊆ ∪ ran ∞Met
2	1	sseli 3972	. 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷 ∈ ∪ ran ∞Met)
3		mopnval.1	. . 3 ⊢ 𝐽 = (MetOpen‘𝐷)
4		fveq2 6896	. . . . . 6 ⊢ (𝑑 = 𝐷 → (ball‘𝑑) = (ball‘𝐷))
5	4	rneqd 5940	. . . . 5 ⊢ (𝑑 = 𝐷 → ran (ball‘𝑑) = ran (ball‘𝐷))
6	5	fveq2d 6900	. . . 4 ⊢ (𝑑 = 𝐷 → (topGen‘ran (ball‘𝑑)) = (topGen‘ran (ball‘𝐷)))
7		df-mopn 21292	. . . 4 ⊢ MetOpen = (𝑑 ∈ ∪ ran ∞Met ↦ (topGen‘ran (ball‘𝑑)))
8		fvex 6909	. . . 4 ⊢ (topGen‘ran (ball‘𝐷)) ∈ V
9	6, 7, 8	fvmpt 7004	. . 3 ⊢ (𝐷 ∈ ∪ ran ∞Met → (MetOpen‘𝐷) = (topGen‘ran (ball‘𝐷)))
10	3, 9	eqtrid 2777	. 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 1533   ∈ wcel 2098  ∪ cuni 4909  ran crn 5679  ‘cfv 6549  topGenctg 17422  ∞Metcxmet 21281  ballcbl 21283  MetOpencmopn 21286

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-iota 6501  df-fun 6551  df-fv 6557  df-mopn 21292

This theorem is referenced by:  mopntopon  24389  elmopn  24392  imasf1oxms  24442  blssopn  24448  metss  24461  prdsxmslem2  24482  metcnp3  24493  xmetutop  24521  tgioo  24756  ismtyhmeolem  37405