Theorem opni 9141

Description: An open set of a metric space includes a ball around each of its points.

Hypothesis

Ref	Expression
opni.1	|- J = (Open` D)

Assertion

Ref	Expression
opni	|- ((D e. Met /\ A e. J /\ P e. A) -> E.x e. ran ( ball ` D)(P e. x /\ x C_ A))

Distinct variable groups:   x,A   x,D   x,P

Proof of Theorem opni

Step	Hyp	Ref	Expression
1		eqid 1884	. . . 4 |- dom dom D = dom dom D
2		opni.1	. . . 4 |- J = (Open` D)
3	1, 2	isopn 9136	. . 3 |- (D e. Met -> (A e. J <-> (A C_ dom dom D /\ A.y e. A E.x e. ran ( ball ` D)(y e. x /\ x C_ A))))
4		eleq1 1957	. . . . . . 7 |- (y = P -> (y e. x <-> P e. x))
5	4	anbi1d 679	. . . . . 6 |- (y = P -> ((y e. x /\ x C_ A) <-> (P e. x /\ x C_ A)))
6	5	rexbidv 2124	. . . . 5 |- (y = P -> (E.x e. ran ( ball ` D)(y e. x /\ x C_ A) <-> E.x e. ran ( ball ` D)(P e. x /\ x C_ A)))
7	6	rcla4cv 2377	. . . 4 |- (A.y e. A E.x e. ran ( ball ` D)(y e. x /\ x C_ A) -> (P e. A -> E.x e. ran ( ball ` D)(P e. x /\ x C_ A)))
8	7	adantl 424	. . 3 |- ((A C_ dom dom D /\ A.y e. A E.x e. ran ( ball ` D)(y e. x /\ x C_ A)) -> (P e. A -> E.x e. ran ( ball ` D)(P e. x /\ x C_ A)))
9	3, 8	syl6bi 231	. 2 |- (D e. Met -> (A e. J -> (P e. A -> E.x e. ran ( ball ` D)(P e. x /\ x C_ A))))
10	9	3imp 1061	1 |- ((D e. Met /\ A e. J /\ P e. A) -> E.x e. ran ( ball ` D)(P e. x /\ x C_ A))

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  A.wral 2105  E.wrex 2106   C_ wss 2593  dom cdm 3986  ran crn 3987  ` cfv 3998  Metcme 9066   ball cbl 9068  Opencopn 9069

This theorem is referenced by:  opni2 9142  opnuni 9145  tgbl 9148  blbas 9149

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-3an 860  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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