Theorem opnfss 7858

Description: The family of open sets of a metric space is a collection of subsets of the base set.

Hypotheses

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

Assertion

Ref	Expression
opnfss	|- (D e. Met -> J (_ P~X)

Proof of Theorem opnfss

Step	Hyp	Ref	Expression
1		opnfval.1	. . 3 |- X = dom dom D
2		opnfval.2	. . 3 |- J = (Open` D)
3	1, 2	opnfval 7857	. 2 |- (D e. Met -> J = {x | (x (_ X /\ A.y e. x E.z e. ran ( ball ` D)(y e. z /\ z (_ x))})
4		pm3.26 319	. . . . 5 |- ((x (_ X /\ A.y e. x E.z e. ran ( ball ` D)(y e. z /\ z (_ x)) -> x (_ X)
5	4	ss2abi 2120	. . . 4 |- {x | (x (_ X /\ A.y e. x E.z e. ran ( ball ` D)(y e. z /\ z (_ x))} (_ {x | x (_ X}
6		df-pw 2402	. . . 4 |- P~X = {x | x (_ X}
7	5, 6	sseqtr4 2094	. . 3 |- {x | (x (_ X /\ A.y e. x E.z e. ran ( ball ` D)(y e. z /\ z (_ x))} (_ P~X
8	7	a1i 8	. 2 |- (D e. Met -> {x | (x (_ X /\ A.y e. x E.z e. ran ( ball ` D)(y e. z /\ z (_ x))} (_ P~X)
9	3, 8	eqsstrd 2095	1 |- (D e. Met -> J (_ P~X)

Syntax hints:   -> wi 3   /\ wa 223   = wceq 956   e. wcel 958  {cab 1463  A.wral 1645  E.wrex 1646   (_ wss 2047  P~cpw 2401  dom cdm 3170  ran crn 3171  ` cfv 3182  Metcme 7789   ball cbl 7791  Opencopn 7792

This theorem is referenced by:  uniopn 7861  opnuni 7868

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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