Theorem opnfval 9134

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 (Open` D) is the family of all open sets in the metric space determined by the metric D. By opntop 9147, the open sets of a metric space form a topology J, whose base set is U.J by uniopn2 9138.

Hypotheses

Ref	Expression
opnfval.1	|- X = dom dom D
opnfval.2	|- J = (Open` D)

Assertion

Ref	Expression
opnfval	|- (D e. Met -> J = {x | (x C_ X /\ A.y e. x E.z e. ran ( ball ` D)(y e. z /\ z C_ x))})

Distinct variable groups:   x,y,z,D   x,X,y,z

Proof of Theorem opnfval

Step	Hyp	Ref	Expression
1		dmexg 4206	. . . . . 6 |- (D e. Met -> dom D e. _V)
2		dmexg 4206	. . . . . 6 |- (dom D e. _V -> dom dom D e. _V)
3	1, 2	syl 12	. . . . 5 |- (D e. Met -> dom dom D e. _V)
4		opnfval.1	. . . . 5 |- X = dom dom D
5	3, 4	syl5eqel 1975	. . . 4 |- (D e. Met -> X e. _V)
6		abssexg 3490	. . . 4 |- (X e. _V -> {x | (x C_ X /\ A.y e. x E.z e. ran ( ball ` D)(y e. z /\ z C_ x))} e. _V)
7	5, 6	syl 12	. . 3 |- (D e. Met -> {x | (x C_ X /\ A.y e. x E.z e. ran ( ball ` D)(y e. z /\ z C_ x))} e. _V)
8		dmeq 4157	. . . . . . . . 9 |- (w = D -> dom w = dom D)
9	8	dmeqd 4159	. . . . . . . 8 |- (w = D -> dom dom w = dom dom D)
10	9, 4	syl6eqr 1946	. . . . . . 7 |- (w = D -> dom dom w = X)
11	10	sseq2d 2645	. . . . . 6 |- (w = D -> (x C_ dom dom w <-> x C_ X))
12		fveq2 4681	. . . . . . . . 9 |- (w = D -> ( ball ` w) = ( ball ` D))
13	12	rneqd 4188	. . . . . . . 8 |- (w = D -> ran ( ball ` w) = ran ( ball ` D))
14	13	rexeqdv 2270	. . . . . . 7 |- (w = D -> (E.z e. ran ( ball ` w)(y e. z /\ z C_ x) <-> E.z e. ran ( ball ` D)(y e. z /\ z C_ x)))
15	14	ralbidv 2123	. . . . . 6 |- (w = D -> (A.y e. x E.z e. ran ( ball ` w)(y e. z /\ z C_ x) <-> A.y e. x E.z e. ran ( ball ` D)(y e. z /\ z C_ x)))
16	11, 15	anbi12d 690	. . . . 5 |- (w = D -> ((x C_ dom dom w /\ A.y e. x E.z e. ran ( ball ` w)(y e. z /\ z C_ x)) <-> (x C_ X /\ A.y e. x E.z e. ran ( ball ` D)(y e. z /\ z C_ x))))
17	16	abbidv 2008	. . . 4 |- (w = D -> {x | (x C_ dom dom w /\ A.y e. x E.z e. ran ( ball ` w)(y e. z /\ z C_ x))} = {x | (x C_ X /\ A.y e. x E.z e. ran ( ball ` D)(y e. z /\ z C_ x))})
18		df-opn 9073	. . . 4 |- Open = {<.w, v>. | (w e. Met /\ v = {x | (x C_ dom dom w /\ A.y e. x E.z e. ran ( ball ` w)(y e. z /\ z C_ x))})}
19	17, 18	fvopab4g 4742	. . 3 |- ((D e. Met /\ {x | (x C_ X /\ A.y e. x E.z e. ran ( ball ` D)(y e. z /\ z C_ x))} e. _V) -> (Open` D) = {x | (x C_ X /\ A.y e. x E.z e. ran ( ball ` D)(y e. z /\ z C_ x))})
20	7, 19	mpdan 768	. 2 |- (D e. Met -> (Open` D) = {x | (x C_ X /\ A.y e. x E.z e. ran ( ball ` D)(y e. z /\ z C_ x))})
21		opnfval.2	. 2 |- J = (Open` D)
22	20, 21	syl5eq 1940	1 |- (D e. Met -> J = {x | (x C_ X /\ A.y e. x E.z e. ran ( ball ` D)(y e. z /\ z C_ x))})

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  {cab 1871  A.wral 2105  E.wrex 2106  _Vcvv 2292   C_ wss 2593  dom cdm 3986  ran crn 3987  ` cfv 3998  Metcme 9066   ball cbl 9068  Opencopn 9069

This theorem is referenced by:  opnfss 9135  isopn 9136

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fv 4014  df-opn 9073

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