MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  hauspwpwdom Structured version   Unicode version

Theorem hauspwpwdom 19576
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 5716 . . 3  |-  ( ( cls `  J ) `
 A )  e. 
_V
21a1i 11 . 2  |-  ( ( J  e.  Haus  /\  A  C_  X )  ->  (
( cls `  J
) `  A )  e.  _V )
3 haustop 18950 . . . . . 6  |-  ( J  e.  Haus  ->  J  e. 
Top )
4 hauspwpwf1.x . . . . . . 7  |-  X  = 
U. J
54topopn 18534 . . . . . 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 4454 . . 3  |-  ( ( J  e.  Haus  /\  A  C_  X )  ->  A  e.  _V )
10 pwexg 4491 . . 3  |-  ( A  e.  _V  ->  ~P A  e.  _V )
11 pwexg 4491 . . 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 2443 . . 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 19575 . 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 7342 . 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 1218 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 1369    e. wcel 1756   {cab 2429   E.wrex 2731   _Vcvv 2987    i^i cin 3342    C_ wss 3343   ~Pcpw 3875   U.cuni 4106   class class class wbr 4307    e. cmpt 4365   -1-1->wf1 5430   ` cfv 5433    ~<_ cdom 7323   Topctop 18513   clsccl 18637   Hauscha 18927
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4418  ax-sep 4428  ax-nul 4436  ax-pow 4485  ax-pr 4546  ax-un 6387
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2735  df-rex 2736  df-reu 2737  df-rab 2739  df-v 2989  df-sbc 3202  df-csb 3304  df-dif 3346  df-un 3348  df-in 3350  df-ss 3357  df-nul 3653  df-if 3807  df-pw 3877  df-sn 3893  df-pr 3895  df-op 3899  df-uni 4107  df-int 4144  df-iun 4188  df-iin 4189  df-br 4308  df-opab 4366  df-mpt 4367  df-id 4651  df-xp 4861  df-rel 4862  df-cnv 4863  df-co 4864  df-dm 4865  df-rn 4866  df-res 4867  df-ima 4868  df-iota 5396  df-fun 5435  df-fn 5436  df-f 5437  df-f1 5438  df-fo 5439  df-f1o 5440  df-fv 5441  df-dom 7327  df-top 18518  df-cld 18638  df-ntr 18639  df-cls 18640  df-haus 18934
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator