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Theorem hmphdis 17749
Description: Homeomorphisms preserve topological discretion. (Contributed by Mario Carneiro, 10-Sep-2015.)
Hypothesis
Ref Expression
hmphdis.1  |-  X  = 
U. J
Assertion
Ref Expression
hmphdis  |-  ( J  ~=  ~P A  ->  J  =  ~P X
)

Proof of Theorem hmphdis
Dummy variables  x  f are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pwuni 4336 . . . 4  |-  J  C_  ~P U. J
2 hmphdis.1 . . . . 5  |-  X  = 
U. J
32pweqi 3746 . . . 4  |-  ~P X  =  ~P U. J
41, 3sseqtr4i 3324 . . 3  |-  J  C_  ~P X
54a1i 11 . 2  |-  ( J  ~=  ~P A  ->  J  C_  ~P X )
6 hmph 17729 . . 3  |-  ( J  ~=  ~P A  <->  ( J  Homeo  ~P A )  =/=  (/) )
7 n0 3580 . . . 4  |-  ( ( J  Homeo  ~P A
)  =/=  (/)  <->  E. f 
f  e.  ( J 
Homeo  ~P A ) )
8 elpwi 3750 . . . . . . 7  |-  ( x  e.  ~P X  ->  x  C_  X )
9 imassrn 5156 . . . . . . . . . . 11  |-  ( f
" x )  C_  ran  f
10 unipw 4355 . . . . . . . . . . . . . . 15  |-  U. ~P A  =  A
1110eqcomi 2391 . . . . . . . . . . . . . 14  |-  A  = 
U. ~P A
122, 11hmeof1o 17717 . . . . . . . . . . . . 13  |-  ( f  e.  ( J  Homeo  ~P A )  ->  f : X -1-1-onto-> A )
13 f1of 5614 . . . . . . . . . . . . 13  |-  ( f : X -1-1-onto-> A  ->  f : X
--> A )
14 frn 5537 . . . . . . . . . . . . 13  |-  ( f : X --> A  ->  ran  f  C_  A )
1512, 13, 143syl 19 . . . . . . . . . . . 12  |-  ( f  e.  ( J  Homeo  ~P A )  ->  ran  f  C_  A )
1615adantr 452 . . . . . . . . . . 11  |-  ( ( f  e.  ( J 
Homeo  ~P A )  /\  x  C_  X )  ->  ran  f  C_  A )
179, 16syl5ss 3302 . . . . . . . . . 10  |-  ( ( f  e.  ( J 
Homeo  ~P A )  /\  x  C_  X )  -> 
( f " x
)  C_  A )
18 vex 2902 . . . . . . . . . . . 12  |-  f  e. 
_V
19 imaexg 5157 . . . . . . . . . . . 12  |-  ( f  e.  _V  ->  (
f " x )  e.  _V )
2018, 19ax-mp 8 . . . . . . . . . . 11  |-  ( f
" x )  e. 
_V
2120elpw 3748 . . . . . . . . . 10  |-  ( ( f " x )  e.  ~P A  <->  ( f " x )  C_  A )
2217, 21sylibr 204 . . . . . . . . 9  |-  ( ( f  e.  ( J 
Homeo  ~P A )  /\  x  C_  X )  -> 
( f " x
)  e.  ~P A
)
232hmeoopn 17719 . . . . . . . . 9  |-  ( ( f  e.  ( J 
Homeo  ~P A )  /\  x  C_  X )  -> 
( x  e.  J  <->  ( f " x )  e.  ~P A ) )
2422, 23mpbird 224 . . . . . . . 8  |-  ( ( f  e.  ( J 
Homeo  ~P A )  /\  x  C_  X )  ->  x  e.  J )
2524ex 424 . . . . . . 7  |-  ( f  e.  ( J  Homeo  ~P A )  ->  (
x  C_  X  ->  x  e.  J ) )
268, 25syl5 30 . . . . . 6  |-  ( f  e.  ( J  Homeo  ~P A )  ->  (
x  e.  ~P X  ->  x  e.  J ) )
2726ssrdv 3297 . . . . 5  |-  ( f  e.  ( J  Homeo  ~P A )  ->  ~P X  C_  J )
2827exlimiv 1641 . . . 4  |-  ( E. f  f  e.  ( J  Homeo  ~P A
)  ->  ~P X  C_  J )
297, 28sylbi 188 . . 3  |-  ( ( J  Homeo  ~P A
)  =/=  (/)  ->  ~P X  C_  J )
306, 29sylbi 188 . 2  |-  ( J  ~=  ~P A  ->  ~P X  C_  J )
315, 30eqssd 3308 1  |-  ( J  ~=  ~P A  ->  J  =  ~P X
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1717    =/= wne 2550   _Vcvv 2899    C_ wss 3263   (/)c0 3571   ~Pcpw 3742   U.cuni 3957   class class class wbr 4153   ran crn 4819   "cima 4821   -->wf 5390   -1-1-onto->wf1o 5393  (class class class)co 6020    Homeo chmeo 17706    ~= chmph 17707
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-suc 4528  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-1o 6660  df-map 6956  df-top 16886  df-topon 16889  df-cn 17213  df-hmeo 17708  df-hmph 17709
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