Theorem mopnval 24326

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 24328, the open sets of a metric space form a topology 𝐽, whose base set is ∪ 𝐽 by mopnuni 24329. (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 6891	. . 3 ⊢ (∞Met‘𝑋) ⊆ ∪ ran ∞Met
2	1	sseli 3942	. 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷 ∈ ∪ ran ∞Met)
3		mopnval.1	. . 3 ⊢ 𝐽 = (MetOpen‘𝐷)
4		fveq2 6858	. . . . . 6 ⊢ (𝑑 = 𝐷 → (ball‘𝑑) = (ball‘𝐷))
5	4	rneqd 5902	. . . . 5 ⊢ (𝑑 = 𝐷 → ran (ball‘𝑑) = ran (ball‘𝐷))
6	5	fveq2d 6862	. . . 4 ⊢ (𝑑 = 𝐷 → (topGen‘ran (ball‘𝑑)) = (topGen‘ran (ball‘𝐷)))
7		df-mopn 21260	. . . 4 ⊢ MetOpen = (𝑑 ∈ ∪ ran ∞Met ↦ (topGen‘ran (ball‘𝑑)))
8		fvex 6871	. . . 4 ⊢ (topGen‘ran (ball‘𝐷)) ∈ V
9	6, 7, 8	fvmpt 6968	. . 3 ⊢ (𝐷 ∈ ∪ ran ∞Met → (MetOpen‘𝐷) = (topGen‘ran (ball‘𝐷)))
10	3, 9	eqtrid 2776	. 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 1540   ∈ wcel 2109  ∪ cuni 4871  ran crn 5639  ‘cfv 6511  topGenctg 17400  ∞Metcxmet 21249  ballcbl 21251  MetOpencmopn 21254

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-iota 6464  df-fun 6513  df-fv 6519  df-mopn 21260

This theorem is referenced by:  mopntopon  24327  elmopn  24330  imasf1oxms  24377  blssopn  24383  metss  24396  prdsxmslem2  24417  metcnp3  24428  xmetutop  24456  tgioo  24684  ismtyhmeolem  37798