Theorem mopnval 24382

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 24384, the open sets of a metric space form a topology 𝐽, whose base set is ∪ 𝐽 by mopnuni 24385. (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 6865	. . 3 ⊢ (∞Met‘𝑋) ⊆ ∪ ran ∞Met
2	1	sseli 3929	. 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷 ∈ ∪ ran ∞Met)
3		mopnval.1	. . 3 ⊢ 𝐽 = (MetOpen‘𝐷)
4		fveq2 6834	. . . . . 6 ⊢ (𝑑 = 𝐷 → (ball‘𝑑) = (ball‘𝐷))
5	4	rneqd 5887	. . . . 5 ⊢ (𝑑 = 𝐷 → ran (ball‘𝑑) = ran (ball‘𝐷))
6	5	fveq2d 6838	. . . 4 ⊢ (𝑑 = 𝐷 → (topGen‘ran (ball‘𝑑)) = (topGen‘ran (ball‘𝐷)))
7		df-mopn 21305	. . . 4 ⊢ MetOpen = (𝑑 ∈ ∪ ran ∞Met ↦ (topGen‘ran (ball‘𝑑)))
8		fvex 6847	. . . 4 ⊢ (topGen‘ran (ball‘𝐷)) ∈ V
9	6, 7, 8	fvmpt 6941	. . 3 ⊢ (𝐷 ∈ ∪ ran ∞Met → (MetOpen‘𝐷) = (topGen‘ran (ball‘𝐷)))
10	3, 9	eqtrid 2783	. 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 1541   ∈ wcel 2113  ∪ cuni 4863  ran crn 5625  ‘cfv 6492  topGenctg 17357  ∞Metcxmet 21294  ballcbl 21296  MetOpencmopn 21299

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-iota 6448  df-fun 6494  df-fv 6500  df-mopn 21305

This theorem is referenced by:  mopntopon  24383  elmopn  24386  imasf1oxms  24433  blssopn  24439  metss  24452  prdsxmslem2  24473  metcnp3  24484  xmetutop  24512  tgioo  24740  ismtyhmeolem  38005