Theorem opni 7864

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 (_ A))

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

Proof of Theorem opni

Step	Hyp	Ref	Expression
1		eqid 1475	. . . 4 |- dom dom D = dom dom D
2		opni.1	. . . 4 |- J = (Open` D)
3	1, 2	isopn 7859	. . 3 |- (D e. Met -> (A e. J <-> (A (_ dom dom D /\ A.y e. A E.x e. ran ( ball ` D)(y e. x /\ x (_ A))))
4		eleq1 1534	. . . . . . 7 |- (y = P -> (y e. x <-> P e. x))
5	4	anbi1d 617	. . . . . 6 |- (y = P -> ((y e. x /\ x (_ A) <-> (P e. x /\ x (_ A)))
6	5	rexbidv 1664	. . . . 5 |- (y = P -> (E.x e. ran ( ball ` D)(y e. x /\ x (_ A) <-> E.x e. ran ( ball ` D)(P e. x /\ x (_ A)))
7	6	rcla4cv 1874	. . . 4 |- (A.y e. A E.x e. ran ( ball ` D)(y e. x /\ x (_ A) -> (P e. A -> E.x e. ran ( ball ` D)(P e. x /\ x (_ A)))
8	7	adantl 388	. . 3 |- ((A (_ dom dom D /\ A.y e. A E.x e. ran ( ball ` D)(y e. x /\ x (_ A)) -> (P e. A -> E.x e. ran ( ball ` D)(P e. x /\ x (_ A)))
9	3, 8	syl6bi 214	. 2 |- (D e. Met -> (A e. J -> (P e. A -> E.x e. ran ( ball ` D)(P e. x /\ x (_ A))))
10	9	3imp 827	1 |- ((D e. Met /\ A e. J /\ P e. A) -> E.x e. ran ( ball ` D)(P e. x /\ x (_ A))

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 775   = wceq 956   e. wcel 958  A.wral 1645  E.wrex 1646   (_ wss 2047  dom cdm 3170  ran crn 3171  ` cfv 3182  Metcme 7789   ball cbl 7791  Opencopn 7792

This theorem is referenced by:  opni2 7865  opnuni 7868  tgbl 7871  blbas 7872

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-3an 777  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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