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Theorem hauspwpwdom 17942
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 5683 . . 3  |-  ( ( cls `  J ) `
 A )  e. 
_V
21a1i 11 . 2  |-  ( ( J  e.  Haus  /\  A  C_  X )  ->  (
( cls `  J
) `  A )  e.  _V )
3 haustop 17318 . . . . . 6  |-  ( J  e.  Haus  ->  J  e. 
Top )
4 hauspwpwf1.x . . . . . . 7  |-  X  = 
U. J
54topopn 16903 . . . . . 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 4292 . . 3  |-  ( ( J  e.  Haus  /\  A  C_  X )  ->  A  e.  _V )
10 pwexg 4325 . . 3  |-  ( A  e.  _V  ->  ~P A  e.  _V )
11 pwexg 4325 . . 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 2388 . . 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 17941 . 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 7062 . 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 1649    e. wcel 1717   {cab 2374   E.wrex 2651   _Vcvv 2900    i^i cin 3263    C_ wss 3264   ~Pcpw 3743   U.cuni 3958   class class class wbr 4154    e. cmpt 4208   -1-1->wf1 5392   ` cfv 5395    ~<_ cdom 7044   Topctop 16882   clsccl 17006   Hauscha 17295
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-reu 2657  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-int 3994  df-iun 4038  df-iin 4039  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-dom 7048  df-top 16887  df-cld 17007  df-ntr 17008  df-cls 17009  df-haus 17302
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