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Theorem indishmph 17783
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 7076 . 2  |-  ( A 
~~  B  <->  E. f 
f : A -1-1-onto-> B )
2 f1of 5633 . . . . . . 7  |-  ( f : A -1-1-onto-> B  ->  f : A
--> B )
3 f1odm 5637 . . . . . . . . . 10  |-  ( f : A -1-1-onto-> B  ->  dom  f  =  A )
4 vex 2919 . . . . . . . . . . 11  |-  f  e. 
_V
54dmex 5091 . . . . . . . . . 10  |-  dom  f  e.  _V
63, 5syl6eqelr 2493 . . . . . . . . 9  |-  ( f : A -1-1-onto-> B  ->  A  e.  _V )
7 f1ofo 5640 . . . . . . . . 9  |-  ( f : A -1-1-onto-> B  ->  f : A -onto-> B )
8 fornex 5929 . . . . . . . . 9  |-  ( A  e.  _V  ->  (
f : A -onto-> B  ->  B  e.  _V )
)
96, 7, 8sylc 58 . . . . . . . 8  |-  ( f : A -1-1-onto-> B  ->  B  e.  _V )
10 elmapg 6990 . . . . . . . 8  |-  ( ( B  e.  _V  /\  A  e.  _V )  ->  ( f  e.  ( B  ^m  A )  <-> 
f : A --> B ) )
119, 6, 10syl2anc 643 . . . . . . 7  |-  ( f : A -1-1-onto-> B  ->  ( f  e.  ( B  ^m  A
)  <->  f : A --> B ) )
122, 11mpbird 224 . . . . . 6  |-  ( f : A -1-1-onto-> B  ->  f  e.  ( B  ^m  A ) )
13 indistopon 17020 . . . . . . . 8  |-  ( A  e.  _V  ->  { (/) ,  A }  e.  (TopOn `  A ) )
146, 13syl 16 . . . . . . 7  |-  ( f : A -1-1-onto-> B  ->  { (/) ,  A }  e.  (TopOn `  A
) )
15 cnindis 17310 . . . . . . 7  |-  ( ( { (/) ,  A }  e.  (TopOn `  A )  /\  B  e.  _V )  ->  ( { (/) ,  A }  Cn  { (/)
,  B } )  =  ( B  ^m  A ) )
1614, 9, 15syl2anc 643 . . . . . 6  |-  ( f : A -1-1-onto-> B  ->  ( { (/)
,  A }  Cn  {
(/) ,  B }
)  =  ( B  ^m  A ) )
1712, 16eleqtrrd 2481 . . . . 5  |-  ( f : A -1-1-onto-> B  ->  f  e.  ( { (/) ,  A }  Cn  { (/) ,  B }
) )
18 f1ocnv 5646 . . . . . . . 8  |-  ( f : A -1-1-onto-> B  ->  `' f : B -1-1-onto-> A )
19 f1of 5633 . . . . . . . 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 6990 . . . . . . . 8  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( `' f  e.  ( A  ^m  B
)  <->  `' f : B --> A ) )
226, 9, 21syl2anc 643 . . . . . . 7  |-  ( f : A -1-1-onto-> B  ->  ( `' f  e.  ( A  ^m  B )  <->  `' f : B --> A ) )
2320, 22mpbird 224 . . . . . 6  |-  ( f : A -1-1-onto-> B  ->  `' f  e.  ( A  ^m  B
) )
24 indistopon 17020 . . . . . . . 8  |-  ( B  e.  _V  ->  { (/) ,  B }  e.  (TopOn `  B ) )
259, 24syl 16 . . . . . . 7  |-  ( f : A -1-1-onto-> B  ->  { (/) ,  B }  e.  (TopOn `  B
) )
26 cnindis 17310 . . . . . . 7  |-  ( ( { (/) ,  B }  e.  (TopOn `  B )  /\  A  e.  _V )  ->  ( { (/) ,  B }  Cn  { (/)
,  A } )  =  ( A  ^m  B ) )
2725, 6, 26syl2anc 643 . . . . . 6  |-  ( f : A -1-1-onto-> B  ->  ( { (/)
,  B }  Cn  {
(/) ,  A }
)  =  ( A  ^m  B ) )
2823, 27eleqtrrd 2481 . . . . 5  |-  ( f : A -1-1-onto-> B  ->  `' f  e.  ( { (/) ,  B }  Cn  { (/) ,  A } ) )
29 ishmeo 17744 . . . . 5  |-  ( f  e.  ( { (/) ,  A }  Homeo  { (/) ,  B } )  <->  ( f  e.  ( { (/) ,  A }  Cn  { (/) ,  B } )  /\  `' f  e.  ( { (/)
,  B }  Cn  {
(/) ,  A }
) ) )
3017, 28, 29sylanbrc 646 . . . 4  |-  ( f : A -1-1-onto-> B  ->  f  e.  ( { (/) ,  A }  Homeo  { (/) ,  B }
) )
31 hmphi 17762 . . . 4  |-  ( f  e.  ( { (/) ,  A }  Homeo  { (/) ,  B } )  ->  { (/) ,  A }  ~=  { (/) ,  B }
)
3230, 31syl 16 . . 3  |-  ( f : A -1-1-onto-> B  ->  { (/) ,  A }  ~=  { (/) ,  B } )
3332exlimiv 1641 . 2  |-  ( E. f  f : A -1-1-onto-> B  ->  { (/) ,  A }  ~=  { (/) ,  B }
)
341, 33sylbi 188 1  |-  ( A 
~~  B  ->  { (/) ,  A }  ~=  { (/)
,  B } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177   E.wex 1547    = wceq 1649    e. wcel 1721   _Vcvv 2916   (/)c0 3588   {cpr 3775   class class class wbr 4172   `'ccnv 4836   dom cdm 4837   -->wf 5409   -onto->wfo 5411   -1-1-onto->wf1o 5412   ` cfv 5413  (class class class)co 6040    ^m cmap 6977    ~~ cen 7065  TopOnctopon 16914    Cn ccn 17242    Homeo chmeo 17738    ~= chmph 17739
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-suc 4547  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-1o 6683  df-map 6979  df-en 7069  df-top 16918  df-topon 16921  df-cn 17245  df-hmeo 17740  df-hmph 17741
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