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Theorem indishmph 20034
Description: Equinumerous sets equipped with their indiscrete topologies are homeomorphic (which means in that particular case that a segment is homeomorphic to a circle contrary to what Wikipedia claims). (Contributed by FL, 17-Aug-2008.) (Proof shortened by Mario Carneiro, 10-Sep-2015.)
Assertion
Ref Expression
indishmph  |-  ( A 
~~  B  ->  { (/) ,  A }  ~=  { (/)
,  B } )

Proof of Theorem indishmph
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 bren 7522 . 2  |-  ( A 
~~  B  <->  E. f 
f : A -1-1-onto-> B )
2 f1of 5814 . . . . . . 7  |-  ( f : A -1-1-onto-> B  ->  f : A
--> B )
3 f1odm 5818 . . . . . . . . . 10  |-  ( f : A -1-1-onto-> B  ->  dom  f  =  A )
4 vex 3116 . . . . . . . . . . 11  |-  f  e. 
_V
54dmex 6714 . . . . . . . . . 10  |-  dom  f  e.  _V
63, 5syl6eqelr 2564 . . . . . . . . 9  |-  ( f : A -1-1-onto-> B  ->  A  e.  _V )
7 f1ofo 5821 . . . . . . . . 9  |-  ( f : A -1-1-onto-> B  ->  f : A -onto-> B )
8 fornex 6750 . . . . . . . . 9  |-  ( A  e.  _V  ->  (
f : A -onto-> B  ->  B  e.  _V )
)
96, 7, 8sylc 60 . . . . . . . 8  |-  ( f : A -1-1-onto-> B  ->  B  e.  _V )
10 elmapg 7430 . . . . . . . 8  |-  ( ( B  e.  _V  /\  A  e.  _V )  ->  ( f  e.  ( B  ^m  A )  <-> 
f : A --> B ) )
119, 6, 10syl2anc 661 . . . . . . 7  |-  ( f : A -1-1-onto-> B  ->  ( f  e.  ( B  ^m  A
)  <->  f : A --> B ) )
122, 11mpbird 232 . . . . . 6  |-  ( f : A -1-1-onto-> B  ->  f  e.  ( B  ^m  A ) )
13 indistopon 19268 . . . . . . . 8  |-  ( A  e.  _V  ->  { (/) ,  A }  e.  (TopOn `  A ) )
146, 13syl 16 . . . . . . 7  |-  ( f : A -1-1-onto-> B  ->  { (/) ,  A }  e.  (TopOn `  A
) )
15 cnindis 19559 . . . . . . 7  |-  ( ( { (/) ,  A }  e.  (TopOn `  A )  /\  B  e.  _V )  ->  ( { (/) ,  A }  Cn  { (/)
,  B } )  =  ( B  ^m  A ) )
1614, 9, 15syl2anc 661 . . . . . 6  |-  ( f : A -1-1-onto-> B  ->  ( { (/)
,  A }  Cn  {
(/) ,  B }
)  =  ( B  ^m  A ) )
1712, 16eleqtrrd 2558 . . . . 5  |-  ( f : A -1-1-onto-> B  ->  f  e.  ( { (/) ,  A }  Cn  { (/) ,  B }
) )
18 f1ocnv 5826 . . . . . . . 8  |-  ( f : A -1-1-onto-> B  ->  `' f : B -1-1-onto-> A )
19 f1of 5814 . . . . . . . 8  |-  ( `' f : B -1-1-onto-> A  ->  `' f : B --> A )
2018, 19syl 16 . . . . . . 7  |-  ( f : A -1-1-onto-> B  ->  `' f : B --> A )
21 elmapg 7430 . . . . . . . 8  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( `' f  e.  ( A  ^m  B
)  <->  `' f : B --> A ) )
226, 9, 21syl2anc 661 . . . . . . 7  |-  ( f : A -1-1-onto-> B  ->  ( `' f  e.  ( A  ^m  B )  <->  `' f : B --> A ) )
2320, 22mpbird 232 . . . . . 6  |-  ( f : A -1-1-onto-> B  ->  `' f  e.  ( A  ^m  B
) )
24 indistopon 19268 . . . . . . . 8  |-  ( B  e.  _V  ->  { (/) ,  B }  e.  (TopOn `  B ) )
259, 24syl 16 . . . . . . 7  |-  ( f : A -1-1-onto-> B  ->  { (/) ,  B }  e.  (TopOn `  B
) )
26 cnindis 19559 . . . . . . 7  |-  ( ( { (/) ,  B }  e.  (TopOn `  B )  /\  A  e.  _V )  ->  ( { (/) ,  B }  Cn  { (/)
,  A } )  =  ( A  ^m  B ) )
2725, 6, 26syl2anc 661 . . . . . 6  |-  ( f : A -1-1-onto-> B  ->  ( { (/)
,  B }  Cn  {
(/) ,  A }
)  =  ( A  ^m  B ) )
2823, 27eleqtrrd 2558 . . . . 5  |-  ( f : A -1-1-onto-> B  ->  `' f  e.  ( { (/) ,  B }  Cn  { (/) ,  A } ) )
29 ishmeo 19995 . . . . 5  |-  ( f  e.  ( { (/) ,  A } Homeo { (/) ,  B } )  <->  ( f  e.  ( { (/) ,  A }  Cn  { (/) ,  B } )  /\  `' f  e.  ( { (/)
,  B }  Cn  {
(/) ,  A }
) ) )
3017, 28, 29sylanbrc 664 . . . 4  |-  ( f : A -1-1-onto-> B  ->  f  e.  ( { (/) ,  A } Homeo { (/) ,  B }
) )
31 hmphi 20013 . . . 4  |-  ( f  e.  ( { (/) ,  A } Homeo { (/) ,  B } )  ->  { (/) ,  A }  ~=  { (/) ,  B }
)
3230, 31syl 16 . . 3  |-  ( f : A -1-1-onto-> B  ->  { (/) ,  A }  ~=  { (/) ,  B } )
3332exlimiv 1698 . 2  |-  ( E. f  f : A -1-1-onto-> B  ->  { (/) ,  A }  ~=  { (/) ,  B }
)
341, 33sylbi 195 1  |-  ( A 
~~  B  ->  { (/) ,  A }  ~=  { (/)
,  B } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1379   E.wex 1596    e. wcel 1767   _Vcvv 3113   (/)c0 3785   {cpr 4029   class class class wbr 4447   `'ccnv 4998   dom cdm 4999   -->wf 5582   -onto->wfo 5584   -1-1-onto->wf1o 5585   ` cfv 5586  (class class class)co 6282    ^m cmap 7417    ~~ cen 7510  TopOnctopon 19162    Cn ccn 19491   Homeochmeo 19989    ~= chmph 19990
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 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
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 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-1st 6781  df-2nd 6782  df-1o 7127  df-map 7419  df-en 7514  df-top 19166  df-topon 19169  df-cn 19494  df-hmeo 19991  df-hmph 19992
This theorem is referenced by: (None)
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