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Theorem hauspwpwdom 17683
Description: If  X is a Hausdorff space, then the cardinality of the closure of a set  A is bounded by the double powerset of  A. In particular, a Hausdorff space with a dense subset  A has cardinality at most  ~P ~P A, and a separable Hausdorff space has cardinality at most  ~P ~P NN. (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Mario Carneiro, 28-Jul-2015.)
Hypothesis
Ref Expression
hauspwpwf1.x  |-  X  = 
U. J
Assertion
Ref Expression
hauspwpwdom  |-  ( ( J  e.  Haus  /\  A  C_  X )  ->  (
( cls `  J
) `  A )  ~<_  ~P ~P A )

Proof of Theorem hauspwpwdom
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fvex 5539 . . 3  |-  ( ( cls `  J ) `
 A )  e. 
_V
21a1i 10 . 2  |-  ( ( J  e.  Haus  /\  A  C_  X )  ->  (
( cls `  J
) `  A )  e.  _V )
3 simpr 447 . . . 4  |-  ( ( J  e.  Haus  /\  A  C_  X )  ->  A  C_  X )
4 haustop 17059 . . . . . 6  |-  ( J  e.  Haus  ->  J  e. 
Top )
5 hauspwpwf1.x . . . . . . 7  |-  X  = 
U. J
65topopn 16652 . . . . . 6  |-  ( J  e.  Top  ->  X  e.  J )
74, 6syl 15 . . . . 5  |-  ( J  e.  Haus  ->  X  e.  J )
87adantr 451 . . . 4  |-  ( ( J  e.  Haus  /\  A  C_  X )  ->  X  e.  J )
9 ssexg 4160 . . . 4  |-  ( ( A  C_  X  /\  X  e.  J )  ->  A  e.  _V )
103, 8, 9syl2anc 642 . . 3  |-  ( ( J  e.  Haus  /\  A  C_  X )  ->  A  e.  _V )
11 pwexg 4194 . . 3  |-  ( A  e.  _V  ->  ~P A  e.  _V )
12 pwexg 4194 . . 3  |-  ( ~P A  e.  _V  ->  ~P ~P A  e.  _V )
1310, 11, 123syl 18 . 2  |-  ( ( J  e.  Haus  /\  A  C_  X )  ->  ~P ~P A  e.  _V )
14 eqid 2283 . . 3  |-  ( x  e.  ( ( cls `  J ) `  A
)  |->  { z  |  E. y  e.  J  ( x  e.  y  /\  z  =  (
y  i^i  A )
) } )  =  ( x  e.  ( ( cls `  J
) `  A )  |->  { z  |  E. y  e.  J  (
x  e.  y  /\  z  =  ( y  i^i  A ) ) } )
155, 14hauspwpwf1 17682 . 2  |-  ( ( J  e.  Haus  /\  A  C_  X )  ->  (
x  e.  ( ( cls `  J ) `
 A )  |->  { z  |  E. y  e.  J  ( x  e.  y  /\  z  =  ( y  i^i 
A ) ) } ) : ( ( cls `  J ) `
 A ) -1-1-> ~P ~P A )
16 f1dom2g 6879 . 2  |-  ( ( ( ( cls `  J
) `  A )  e.  _V  /\  ~P ~P A  e.  _V  /\  (
x  e.  ( ( cls `  J ) `
 A )  |->  { z  |  E. y  e.  J  ( x  e.  y  /\  z  =  ( y  i^i 
A ) ) } ) : ( ( cls `  J ) `
 A ) -1-1-> ~P ~P A )  ->  (
( cls `  J
) `  A )  ~<_  ~P ~P A )
172, 13, 15, 16syl3anc 1182 1  |-  ( ( J  e.  Haus  /\  A  C_  X )  ->  (
( cls `  J
) `  A )  ~<_  ~P ~P A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   {cab 2269   E.wrex 2544   _Vcvv 2788    i^i cin 3151    C_ wss 3152   ~Pcpw 3625   U.cuni 3827   class class class wbr 4023    e. cmpt 4077   -1-1->wf1 5252   ` cfv 5255    ~<_ cdom 6861   Topctop 16631   clsccl 16755   Hauscha 17036
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-dom 6865  df-top 16636  df-cld 16756  df-ntr 16757  df-cls 16758  df-haus 17043
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