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Theorem hauspwpwdom 20357
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 5882 . . 3  |-  ( ( cls `  J ) `
 A )  e. 
_V
21a1i 11 . 2  |-  ( ( J  e.  Haus  /\  A  C_  X )  ->  (
( cls `  J
) `  A )  e.  _V )
3 haustop 19700 . . . . . 6  |-  ( J  e.  Haus  ->  J  e. 
Top )
4 hauspwpwf1.x . . . . . . 7  |-  X  = 
U. J
54topopn 19284 . . . . . 6  |-  ( J  e.  Top  ->  X  e.  J )
63, 5syl 16 . . . . 5  |-  ( J  e.  Haus  ->  X  e.  J )
76adantr 465 . . . 4  |-  ( ( J  e.  Haus  /\  A  C_  X )  ->  X  e.  J )
8 simpr 461 . . . 4  |-  ( ( J  e.  Haus  /\  A  C_  X )  ->  A  C_  X )
97, 8ssexd 4600 . . 3  |-  ( ( J  e.  Haus  /\  A  C_  X )  ->  A  e.  _V )
10 pwexg 4637 . . 3  |-  ( A  e.  _V  ->  ~P A  e.  _V )
11 pwexg 4637 . . 3  |-  ( ~P A  e.  _V  ->  ~P ~P A  e.  _V )
129, 10, 113syl 20 . 2  |-  ( ( J  e.  Haus  /\  A  C_  X )  ->  ~P ~P A  e.  _V )
13 eqid 2467 . . 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 20356 . 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 7545 . 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 1228 1  |-  ( ( J  e.  Haus  /\  A  C_  X )  ->  (
( cls `  J
) `  A )  ~<_  ~P ~P A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   {cab 2452   E.wrex 2818   _Vcvv 3118    i^i cin 3480    C_ wss 3481   ~Pcpw 4016   U.cuni 4251   class class class wbr 4453    |-> cmpt 4511   -1-1->wf1 5591   ` cfv 5594    ~<_ cdom 7526   Topctop 19263   clsccl 19387   Hauscha 19677
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-iin 4334  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-dom 7530  df-top 19268  df-cld 19388  df-ntr 19389  df-cls 19390  df-haus 19684
This theorem is referenced by: (None)
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