Theorem opnfval 7857

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 7870, the open sets of a metric space form a topology J, whose base set is U.J by uniopn 7861.

Hypotheses

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

Assertion

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

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

Proof of Theorem opnfval

Step	Hyp	Ref	Expression
1		dmexg 3358	. . . . . 6 |- (D e. Met -> dom D e. V)
2		dmexg 3358	. . . . . 6 |- (dom D e. V -> dom dom D e. V)
3	1, 2	syl 10	. . . . 5 |- (D e. Met -> dom dom D e. V)
4		opnfval.1	. . . . 5 |- X = dom dom D
5	3, 4	syl5eqel 1552	. . . 4 |- (D e. Met -> X e. V)
6		abssexg 2747	. . . 4 |- (X e. V -> {x | (x (_ X /\ A.y e. x E.z e. ran ( ball ` D)(y e. z /\ z (_ x))} e. V)
7	5, 6	syl 10	. . 3 |- (D e. Met -> {x | (x (_ X /\ A.y e. x E.z e. ran ( ball ` D)(y e. z /\ z (_ x))} e. V)
8		dmeq 3311	. . . . . . . . 9 |- (w = D -> dom w = dom D)
9	8	dmeqd 3313	. . . . . . . 8 |- (w = D -> dom dom w = dom dom D)
10	9, 4	syl6eqr 1525	. . . . . . 7 |- (w = D -> dom dom w = X)
11	10	sseq2d 2089	. . . . . 6 |- (w = D -> (x (_ dom dom w <-> x (_ X))
12		fveq2 3724	. . . . . . . . 9 |- (w = D -> ( ball ` w) = ( ball ` D))
13	12	rneqd 3341	. . . . . . . 8 |- (w = D -> ran ( ball ` w) = ran ( ball ` D))
14	13	rexeq1d 1790	. . . . . . 7 |- (w = D -> (E.z e. ran ( ball ` w)(y e. z /\ z (_ x) <-> E.z e. ran ( ball ` D)(y e. z /\ z (_ x)))
15	14	ralbidv 1663	. . . . . 6 |- (w = D -> (A.y e. x E.z e. ran ( ball ` w)(y e. z /\ z (_ x) <-> A.y e. x E.z e. ran ( ball ` D)(y e. z /\ z (_ x)))
16	11, 15	anbi12d 628	. . . . 5 |- (w = D -> ((x (_ dom dom w /\ A.y e. x E.z e. ran ( ball ` w)(y e. z /\ z (_ x)) <-> (x (_ X /\ A.y e. x E.z e. ran ( ball ` D)(y e. z /\ z (_ x))))
17	16	abbidv 1577	. . . 4 |- (w = D -> {x | (x (_ dom dom w /\ A.y e. x E.z e. ran ( ball ` w)(y e. z /\ z (_ x))} = {x | (x (_ X /\ A.y e. x E.z e. ran ( ball ` D)(y e. z /\ z (_ x))})
18		df-opn 7796	. . . 4 |- Open = {<.w, v>. | (w e. Met /\ v = {x | (x (_ dom dom w /\ A.y e. x E.z e. ran ( ball ` w)(y e. z /\ z (_ x))})}
19	17, 18	fvopab4g 3779	. . 3 |- ((D e. Met /\ {x | (x (_ X /\ A.y e. x E.z e. ran ( ball ` D)(y e. z /\ z (_ x))} e. V) -> (Open` D) = {x | (x (_ X /\ A.y e. x E.z e. ran ( ball ` D)(y e. z /\ z (_ x))})
20	7, 19	mpdan 704	. 2 |- (D e. Met -> (Open` D) = {x | (x (_ X /\ A.y e. x E.z e. ran ( ball ` D)(y e. z /\ z (_ x))})
21		opnfval.2	. 2 |- J = (Open` D)
22	20, 21	syl5eq 1519	1 |- (D e. Met -> J = {x | (x (_ X /\ A.y e. x E.z e. ran ( ball ` D)(y e. z /\ z (_ x))})

Syntax hints:   -> wi 3   /\ wa 223   = wceq 956   e. wcel 958  {cab 1463  A.wral 1645  E.wrex 1646  Vcvv 1811   (_ wss 2047  dom cdm 3170  ran crn 3171  ` cfv 3182  Metcme 7789   ball cbl 7791  Opencopn 7792

This theorem is referenced by:  opnfss 7858  isopn 7859

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-rab 1652  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fv 3198  df-opn 7796

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