Theorem opnfval 7854

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 7867, the open sets of a metric space form a topology J, whose base set is U.J by uniopn 7858.

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 3364	. . . . . 6 |- (D e. Met -> dom D e. V)
2		dmexg 3364	. . . . . 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 1555	. . . 4 |- (D e. Met -> X e. V)
6		abssexg 2753	. . . 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 3317	. . . . . . . . 9 |- (w = D -> dom w = dom D)
9	8	dmeqd 3319	. . . . . . . 8 |- (w = D -> dom dom w = dom dom D)
10	9, 4	syl6eqr 1528	. . . . . . 7 |- (w = D -> dom dom w = X)
11	10	sseq2d 2092	. . . . . 6 |- (w = D -> (x (_ dom dom w <-> x (_ X))
12		fveq2 3730	. . . . . . . . 9 |- (w = D -> ( ball ` w) = ( ball ` D))
13	12	rneqd 3347	. . . . . . . 8 |- (w = D -> ran ( ball ` w) = ran ( ball ` D))
14	13	rexeq1d 1793	. . . . . . 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 1666	. . . . . 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 630	. . . . 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 1580	. . . 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 7793	. . . 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 3785	. . 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 706	. 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 1522	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 958   e. wcel 960  {cab 1466  A.wral 1648  E.wrex 1649  Vcvv 1814   (_ wss 2050  dom cdm 3176  ran crn 3177  ` cfv 3188  Metcme 7786   ball cbl 7788  Opencopn 7789

This theorem is referenced by:  opnfss 7855  isopn 7856

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-rab 1655  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fv 3204  df-opn 7793

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