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Theorem indishmph 17505
Description: Equinumerous sets equipped with their indiscrete topologies are homeomorph (which means in that particular case that a segment is homeomorph 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 6887 . 2  |-  ( A 
~~  B  <->  E. f 
f : A -1-1-onto-> B )
2 f1of 5488 . . . . . . 7  |-  ( f : A -1-1-onto-> B  ->  f : A
--> B )
3 f1odm 5492 . . . . . . . . . 10  |-  ( f : A -1-1-onto-> B  ->  dom  f  =  A )
4 vex 2804 . . . . . . . . . . 11  |-  f  e. 
_V
54dmex 4957 . . . . . . . . . 10  |-  dom  f  e.  _V
63, 5syl6eqelr 2385 . . . . . . . . 9  |-  ( f : A -1-1-onto-> B  ->  A  e.  _V )
7 f1ofo 5495 . . . . . . . . 9  |-  ( f : A -1-1-onto-> B  ->  f : A -onto-> B )
8 fornex 5766 . . . . . . . . 9  |-  ( A  e.  _V  ->  (
f : A -onto-> B  ->  B  e.  _V )
)
96, 7, 8sylc 56 . . . . . . . 8  |-  ( f : A -1-1-onto-> B  ->  B  e.  _V )
10 elmapg 6801 . . . . . . . 8  |-  ( ( B  e.  _V  /\  A  e.  _V )  ->  ( f  e.  ( B  ^m  A )  <-> 
f : A --> B ) )
119, 6, 10syl2anc 642 . . . . . . 7  |-  ( f : A -1-1-onto-> B  ->  ( f  e.  ( B  ^m  A
)  <->  f : A --> B ) )
122, 11mpbird 223 . . . . . 6  |-  ( f : A -1-1-onto-> B  ->  f  e.  ( B  ^m  A ) )
13 indistopon 16754 . . . . . . . 8  |-  ( A  e.  _V  ->  { (/) ,  A }  e.  (TopOn `  A ) )
146, 13syl 15 . . . . . . 7  |-  ( f : A -1-1-onto-> B  ->  { (/) ,  A }  e.  (TopOn `  A
) )
15 cnindis 17036 . . . . . . 7  |-  ( ( { (/) ,  A }  e.  (TopOn `  A )  /\  B  e.  _V )  ->  ( { (/) ,  A }  Cn  { (/)
,  B } )  =  ( B  ^m  A ) )
1614, 9, 15syl2anc 642 . . . . . 6  |-  ( f : A -1-1-onto-> B  ->  ( { (/)
,  A }  Cn  {
(/) ,  B }
)  =  ( B  ^m  A ) )
1712, 16eleqtrrd 2373 . . . . 5  |-  ( f : A -1-1-onto-> B  ->  f  e.  ( { (/) ,  A }  Cn  { (/) ,  B }
) )
18 f1ocnv 5501 . . . . . . . 8  |-  ( f : A -1-1-onto-> B  ->  `' f : B -1-1-onto-> A )
19 f1of 5488 . . . . . . . 8  |-  ( `' f : B -1-1-onto-> A  ->  `' f : B --> A )
2018, 19syl 15 . . . . . . 7  |-  ( f : A -1-1-onto-> B  ->  `' f : B --> A )
21 elmapg 6801 . . . . . . . 8  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( `' f  e.  ( A  ^m  B
)  <->  `' f : B --> A ) )
226, 9, 21syl2anc 642 . . . . . . 7  |-  ( f : A -1-1-onto-> B  ->  ( `' f  e.  ( A  ^m  B )  <->  `' f : B --> A ) )
2320, 22mpbird 223 . . . . . 6  |-  ( f : A -1-1-onto-> B  ->  `' f  e.  ( A  ^m  B
) )
24 indistopon 16754 . . . . . . . 8  |-  ( B  e.  _V  ->  { (/) ,  B }  e.  (TopOn `  B ) )
259, 24syl 15 . . . . . . 7  |-  ( f : A -1-1-onto-> B  ->  { (/) ,  B }  e.  (TopOn `  B
) )
26 cnindis 17036 . . . . . . 7  |-  ( ( { (/) ,  B }  e.  (TopOn `  B )  /\  A  e.  _V )  ->  ( { (/) ,  B }  Cn  { (/)
,  A } )  =  ( A  ^m  B ) )
2725, 6, 26syl2anc 642 . . . . . 6  |-  ( f : A -1-1-onto-> B  ->  ( { (/)
,  B }  Cn  {
(/) ,  A }
)  =  ( A  ^m  B ) )
2823, 27eleqtrrd 2373 . . . . 5  |-  ( f : A -1-1-onto-> B  ->  `' f  e.  ( { (/) ,  B }  Cn  { (/) ,  A } ) )
29 ishmeo 17466 . . . . 5  |-  ( f  e.  ( { (/) ,  A }  Homeo  { (/) ,  B } )  <->  ( f  e.  ( { (/) ,  A }  Cn  { (/) ,  B } )  /\  `' f  e.  ( { (/)
,  B }  Cn  {
(/) ,  A }
) ) )
3017, 28, 29sylanbrc 645 . . . 4  |-  ( f : A -1-1-onto-> B  ->  f  e.  ( { (/) ,  A }  Homeo  { (/) ,  B }
) )
31 hmphi 17484 . . . 4  |-  ( f  e.  ( { (/) ,  A }  Homeo  { (/) ,  B } )  ->  { (/) ,  A }  ~=  { (/) ,  B }
)
3230, 31syl 15 . . 3  |-  ( f : A -1-1-onto-> B  ->  { (/) ,  A }  ~=  { (/) ,  B } )
3332exlimiv 1624 . 2  |-  ( E. f  f : A -1-1-onto-> B  ->  { (/) ,  A }  ~=  { (/) ,  B }
)
341, 33sylbi 187 1  |-  ( A 
~~  B  ->  { (/) ,  A }  ~=  { (/)
,  B } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   E.wex 1531    = wceq 1632    e. wcel 1696   _Vcvv 2801   (/)c0 3468   {cpr 3654   class class class wbr 4039   `'ccnv 4704   dom cdm 4705   -->wf 5267   -onto->wfo 5269   -1-1-onto->wf1o 5270   ` cfv 5271  (class class class)co 5874    ^m cmap 6788    ~~ cen 6876  TopOnctopon 16648    Cn ccn 16970    Homeo chmeo 17460    ~= chmph 17461
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-suc 4414  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-1o 6495  df-map 6790  df-en 6880  df-top 16652  df-topon 16655  df-cn 16973  df-hmeo 17462  df-hmph 17463
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