Theorem hauspwpwdom 21792

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.

Step	Hyp	Ref	Expression
1		fvexd 6203	. 2 |- ( ( J e. Haus /\ A C_ X ) -> ( ( cls ` J ) ` A ) e. _V )
2		haustop 21135	. . . . . 6 |- ( J e. Haus -> J e. Top )
3		hauspwpwf1.x	. . . . . . 7 |- X = U. J
4	3	topopn 20711	. . . . . 6 |- ( J e. Top -> X e. J )
5	2, 4	syl 17	. . . . 5 |- ( J e. Haus -> X e. J )
6	5	adantr 481	. . . 4 |- ( ( J e. Haus /\ A C_ X ) -> X e. J )
7		simpr 477	. . . 4 |- ( ( J e. Haus /\ A C_ X ) -> A C_ X )
8	6, 7	ssexd 4805	. . 3 |- ( ( J e. Haus /\ A C_ X ) -> A e. _V )
9		pwexg 4850	. . 3 |- ( A e. _V -> ~P A e. _V )
10		pwexg 4850	. . 3 |- ( ~P A e. _V -> ~P ~P A e. _V )
11	8, 9, 10	3syl 18	. 2 |- ( ( J e. Haus /\ A C_ X ) -> ~P ~P A e. _V )
12		eqid 2622	. . 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 ) ) } )
13	3, 12	hauspwpwf1 21791	. 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 )
14		f1dom2g 7973	. 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 )
15	1, 11, 13, 14	syl3anc 1326	1 |- ( ( J e. Haus /\ A C_ X ) -> ( ( cls ` J ) ` A ) ~<_ ~P ~P A )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   E.wrex 2913   _Vcvv 3200   i^i cin 3573   C_ wss 3574   ~Pcpw 4158   U.cuni 4436   class class class wbr 4653   |-> cmpt 4729   -1-1->wf1 5885   ` cfv 5888   ~<_ cdom 7953   Topctop 20698   clsccl 20822   Hauscha 21112

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-dom 7957  df-top 20699  df-cld 20823  df-ntr 20824  df-cls 20825  df-haus 21119

This theorem is referenced by: (None)