MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  indishmph Structured version   Unicode version

Theorem indishmph 20425
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 7544 . 2  |-  ( A 
~~  B  <->  E. f 
f : A -1-1-onto-> B )
2 f1of 5822 . . . . . . 7  |-  ( f : A -1-1-onto-> B  ->  f : A
--> B )
3 f1odm 5826 . . . . . . . . . 10  |-  ( f : A -1-1-onto-> B  ->  dom  f  =  A )
4 vex 3112 . . . . . . . . . . 11  |-  f  e. 
_V
54dmex 6732 . . . . . . . . . 10  |-  dom  f  e.  _V
63, 5syl6eqelr 2554 . . . . . . . . 9  |-  ( f : A -1-1-onto-> B  ->  A  e.  _V )
7 f1ofo 5829 . . . . . . . . 9  |-  ( f : A -1-1-onto-> B  ->  f : A -onto-> B )
8 fornex 6768 . . . . . . . . 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 )
109, 6elmapd 7452 . . . . . . 7  |-  ( f : A -1-1-onto-> B  ->  ( f  e.  ( B  ^m  A
)  <->  f : A --> B ) )
112, 10mpbird 232 . . . . . 6  |-  ( f : A -1-1-onto-> B  ->  f  e.  ( B  ^m  A ) )
12 indistopon 19629 . . . . . . . 8  |-  ( A  e.  _V  ->  { (/) ,  A }  e.  (TopOn `  A ) )
136, 12syl 16 . . . . . . 7  |-  ( f : A -1-1-onto-> B  ->  { (/) ,  A }  e.  (TopOn `  A
) )
14 cnindis 19920 . . . . . . 7  |-  ( ( { (/) ,  A }  e.  (TopOn `  A )  /\  B  e.  _V )  ->  ( { (/) ,  A }  Cn  { (/)
,  B } )  =  ( B  ^m  A ) )
1513, 9, 14syl2anc 661 . . . . . 6  |-  ( f : A -1-1-onto-> B  ->  ( { (/)
,  A }  Cn  {
(/) ,  B }
)  =  ( B  ^m  A ) )
1611, 15eleqtrrd 2548 . . . . 5  |-  ( f : A -1-1-onto-> B  ->  f  e.  ( { (/) ,  A }  Cn  { (/) ,  B }
) )
17 f1ocnv 5834 . . . . . . . 8  |-  ( f : A -1-1-onto-> B  ->  `' f : B -1-1-onto-> A )
18 f1of 5822 . . . . . . . 8  |-  ( `' f : B -1-1-onto-> A  ->  `' f : B --> A )
1917, 18syl 16 . . . . . . 7  |-  ( f : A -1-1-onto-> B  ->  `' f : B --> A )
206, 9elmapd 7452 . . . . . . 7  |-  ( f : A -1-1-onto-> B  ->  ( `' f  e.  ( A  ^m  B )  <->  `' f : B --> A ) )
2119, 20mpbird 232 . . . . . 6  |-  ( f : A -1-1-onto-> B  ->  `' f  e.  ( A  ^m  B
) )
22 indistopon 19629 . . . . . . . 8  |-  ( B  e.  _V  ->  { (/) ,  B }  e.  (TopOn `  B ) )
239, 22syl 16 . . . . . . 7  |-  ( f : A -1-1-onto-> B  ->  { (/) ,  B }  e.  (TopOn `  B
) )
24 cnindis 19920 . . . . . . 7  |-  ( ( { (/) ,  B }  e.  (TopOn `  B )  /\  A  e.  _V )  ->  ( { (/) ,  B }  Cn  { (/)
,  A } )  =  ( A  ^m  B ) )
2523, 6, 24syl2anc 661 . . . . . 6  |-  ( f : A -1-1-onto-> B  ->  ( { (/)
,  B }  Cn  {
(/) ,  A }
)  =  ( A  ^m  B ) )
2621, 25eleqtrrd 2548 . . . . 5  |-  ( f : A -1-1-onto-> B  ->  `' f  e.  ( { (/) ,  B }  Cn  { (/) ,  A } ) )
27 ishmeo 20386 . . . . 5  |-  ( f  e.  ( { (/) ,  A } Homeo { (/) ,  B } )  <->  ( f  e.  ( { (/) ,  A }  Cn  { (/) ,  B } )  /\  `' f  e.  ( { (/)
,  B }  Cn  {
(/) ,  A }
) ) )
2816, 26, 27sylanbrc 664 . . . 4  |-  ( f : A -1-1-onto-> B  ->  f  e.  ( { (/) ,  A } Homeo { (/) ,  B }
) )
29 hmphi 20404 . . . 4  |-  ( f  e.  ( { (/) ,  A } Homeo { (/) ,  B } )  ->  { (/) ,  A }  ~=  { (/) ,  B }
)
3028, 29syl 16 . . 3  |-  ( f : A -1-1-onto-> B  ->  { (/) ,  A }  ~=  { (/) ,  B } )
3130exlimiv 1723 . 2  |-  ( E. f  f : A -1-1-onto-> B  ->  { (/) ,  A }  ~=  { (/) ,  B }
)
321, 31sylbi 195 1  |-  ( A 
~~  B  ->  { (/) ,  A }  ~=  { (/)
,  B } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1395   E.wex 1613    e. wcel 1819   _Vcvv 3109   (/)c0 3793   {cpr 4034   class class class wbr 4456   `'ccnv 5007   dom cdm 5008   -->wf 5590   -onto->wfo 5592   -1-1-onto->wf1o 5593   ` cfv 5594  (class class class)co 6296    ^m cmap 7438    ~~ cen 7532  TopOnctopon 19522    Cn ccn 19852   Homeochmeo 20380    ~= chmph 20381
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-1o 7148  df-map 7440  df-en 7536  df-top 19526  df-topon 19529  df-cn 19855  df-hmeo 20382  df-hmph 20383
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator