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Theorem hauspwpwdom 18012
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 5734 . . 3  |-  ( ( cls `  J ) `
 A )  e. 
_V
21a1i 11 . 2  |-  ( ( J  e.  Haus  /\  A  C_  X )  ->  (
( cls `  J
) `  A )  e.  _V )
3 haustop 17387 . . . . . 6  |-  ( J  e.  Haus  ->  J  e. 
Top )
4 hauspwpwf1.x . . . . . . 7  |-  X  = 
U. J
54topopn 16971 . . . . . 6  |-  ( J  e.  Top  ->  X  e.  J )
63, 5syl 16 . . . . 5  |-  ( J  e.  Haus  ->  X  e.  J )
76adantr 452 . . . 4  |-  ( ( J  e.  Haus  /\  A  C_  X )  ->  X  e.  J )
8 simpr 448 . . . 4  |-  ( ( J  e.  Haus  /\  A  C_  X )  ->  A  C_  X )
97, 8ssexd 4342 . . 3  |-  ( ( J  e.  Haus  /\  A  C_  X )  ->  A  e.  _V )
10 pwexg 4375 . . 3  |-  ( A  e.  _V  ->  ~P A  e.  _V )
11 pwexg 4375 . . 3  |-  ( ~P A  e.  _V  ->  ~P ~P A  e.  _V )
129, 10, 113syl 19 . 2  |-  ( ( J  e.  Haus  /\  A  C_  X )  ->  ~P ~P A  e.  _V )
13 eqid 2435 . . 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 ) ) } )
144, 13hauspwpwf1 18011 . 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 )
15 f1dom2g 7117 . 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 )
162, 12, 14, 15syl3anc 1184 1  |-  ( ( J  e.  Haus  /\  A  C_  X )  ->  (
( cls `  J
) `  A )  ~<_  ~P ~P A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   {cab 2421   E.wrex 2698   _Vcvv 2948    i^i cin 3311    C_ wss 3312   ~Pcpw 3791   U.cuni 4007   class class class wbr 4204    e. cmpt 4258   -1-1->wf1 5443   ` cfv 5446    ~<_ cdom 7099   Topctop 16950   clsccl 17074   Hauscha 17364
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-dom 7103  df-top 16955  df-cld 17075  df-ntr 17076  df-cls 17077  df-haus 17371
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