Theorem mopnval 22053

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 22055, the open sets of a metric space form a topology 𝐽, whose base set is ∪ 𝐽 by mopnuni 22056. (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 6127	. . 3 ⊢ (∞Met‘𝑋) ⊆ ∪ ran ∞Met
2	1	sseli 3564	. 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐷 ∈ ∪ ran ∞Met)
3		mopnval.1	. . 3 ⊢ 𝐽 = (MetOpen‘𝐷)
4		fveq2 6103	. . . . . 6 ⊢ (𝑑 = 𝐷 → (ball‘𝑑) = (ball‘𝐷))
5	4	rneqd 5274	. . . . 5 ⊢ (𝑑 = 𝐷 → ran (ball‘𝑑) = ran (ball‘𝐷))
6	5	fveq2d 6107	. . . 4 ⊢ (𝑑 = 𝐷 → (topGen‘ran (ball‘𝑑)) = (topGen‘ran (ball‘𝐷)))
7		df-mopn 19563	. . . 4 ⊢ MetOpen = (𝑑 ∈ ∪ ran ∞Met ↦ (topGen‘ran (ball‘𝑑)))
8		fvex 6113	. . . 4 ⊢ (topGen‘ran (ball‘𝐷)) ∈ V
9	6, 7, 8	fvmpt 6191	. . 3 ⊢ (𝐷 ∈ ∪ ran ∞Met → (MetOpen‘𝐷) = (topGen‘ran (ball‘𝐷)))
10	3, 9	syl5eq 2656	. 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 1475   ∈ wcel 1977  ∪ cuni 4372  ran crn 5039  ‘cfv 5804  topGenctg 15921  ∞Metcxmt 19552  ballcbl 19554  MetOpencmopn 19557

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fv 5812  df-mopn 19563

This theorem is referenced by:  mopntopon  22054  elmopn  22057  imasf1oxms  22104  blssopn  22110  metss  22123  prdsxmslem2  22144  metcnp3  22155  xmetutop  22183  tgioo  22407  ismtyhmeolem  32773