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Theorem indishmph 17820
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 7109 . 2  |-  ( A 
~~  B  <->  E. f 
f : A -1-1-onto-> B )
2 f1of 5666 . . . . . . 7  |-  ( f : A -1-1-onto-> B  ->  f : A
--> B )
3 f1odm 5670 . . . . . . . . . 10  |-  ( f : A -1-1-onto-> B  ->  dom  f  =  A )
4 vex 2951 . . . . . . . . . . 11  |-  f  e. 
_V
54dmex 5124 . . . . . . . . . 10  |-  dom  f  e.  _V
63, 5syl6eqelr 2524 . . . . . . . . 9  |-  ( f : A -1-1-onto-> B  ->  A  e.  _V )
7 f1ofo 5673 . . . . . . . . 9  |-  ( f : A -1-1-onto-> B  ->  f : A -onto-> B )
8 fornex 5962 . . . . . . . . 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 7023 . . . . . . . 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 17055 . . . . . . . 8  |-  ( A  e.  _V  ->  { (/) ,  A }  e.  (TopOn `  A ) )
146, 13syl 16 . . . . . . 7  |-  ( f : A -1-1-onto-> B  ->  { (/) ,  A }  e.  (TopOn `  A
) )
15 cnindis 17346 . . . . . . 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 2512 . . . . 5  |-  ( f : A -1-1-onto-> B  ->  f  e.  ( { (/) ,  A }  Cn  { (/) ,  B }
) )
18 f1ocnv 5679 . . . . . . . 8  |-  ( f : A -1-1-onto-> B  ->  `' f : B -1-1-onto-> A )
19 f1of 5666 . . . . . . . 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 7023 . . . . . . . 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 17055 . . . . . . . 8  |-  ( B  e.  _V  ->  { (/) ,  B }  e.  (TopOn `  B ) )
259, 24syl 16 . . . . . . 7  |-  ( f : A -1-1-onto-> B  ->  { (/) ,  B }  e.  (TopOn `  B
) )
26 cnindis 17346 . . . . . . 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 2512 . . . . 5  |-  ( f : A -1-1-onto-> B  ->  `' f  e.  ( { (/) ,  B }  Cn  { (/) ,  A } ) )
29 ishmeo 17781 . . . . 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 17799 . . . 4  |-  ( f  e.  ( { (/) ,  A }  Homeo  { (/) ,  B } )  ->  { (/) ,  A }  ~=  { (/) ,  B }
)
3230, 31syl 16 . . 3  |-  ( f : A -1-1-onto-> B  ->  { (/) ,  A }  ~=  { (/) ,  B } )
3332exlimiv 1644 . 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 1550    = wceq 1652    e. wcel 1725   _Vcvv 2948   (/)c0 3620   {cpr 3807   class class class wbr 4204   `'ccnv 4869   dom cdm 4870   -->wf 5442   -onto->wfo 5444   -1-1-onto->wf1o 5445   ` cfv 5446  (class class class)co 6073    ^m cmap 7010    ~~ cen 7098  TopOnctopon 16949    Cn ccn 17278    Homeo chmeo 17775    ~= chmph 17776
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-suc 4579  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-1o 6716  df-map 7012  df-en 7102  df-top 16953  df-topon 16956  df-cn 17281  df-hmeo 17777  df-hmph 17778
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