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Theorem hmphdis 17487
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 4206 . . . 4  |-  J  C_  ~P U. J
2 hmphdis.1 . . . . 5  |-  X  = 
U. J
32pweqi 3629 . . . 4  |-  ~P X  =  ~P U. J
41, 3sseqtr4i 3211 . . 3  |-  J  C_  ~P X
54a1i 10 . 2  |-  ( J  ~=  ~P A  ->  J  C_  ~P X )
6 hmph 17467 . . 3  |-  ( J  ~=  ~P A  <->  ( J  Homeo  ~P A )  =/=  (/) )
7 n0 3464 . . . 4  |-  ( ( J  Homeo  ~P A
)  =/=  (/)  <->  E. f 
f  e.  ( J 
Homeo  ~P A ) )
8 elpwi 3633 . . . . . . 7  |-  ( x  e.  ~P X  ->  x  C_  X )
9 imassrn 5025 . . . . . . . . . . 11  |-  ( f
" x )  C_  ran  f
10 unipw 4224 . . . . . . . . . . . . . . 15  |-  U. ~P A  =  A
1110eqcomi 2287 . . . . . . . . . . . . . 14  |-  A  = 
U. ~P A
122, 11hmeof1o 17455 . . . . . . . . . . . . 13  |-  ( f  e.  ( J  Homeo  ~P A )  ->  f : X -1-1-onto-> A )
13 f1of 5472 . . . . . . . . . . . . 13  |-  ( f : X -1-1-onto-> A  ->  f : X
--> A )
14 frn 5395 . . . . . . . . . . . . 13  |-  ( f : X --> A  ->  ran  f  C_  A )
1512, 13, 143syl 18 . . . . . . . . . . . 12  |-  ( f  e.  ( J  Homeo  ~P A )  ->  ran  f  C_  A )
1615adantr 451 . . . . . . . . . . 11  |-  ( ( f  e.  ( J 
Homeo  ~P A )  /\  x  C_  X )  ->  ran  f  C_  A )
179, 16syl5ss 3190 . . . . . . . . . 10  |-  ( ( f  e.  ( J 
Homeo  ~P A )  /\  x  C_  X )  -> 
( f " x
)  C_  A )
18 vex 2791 . . . . . . . . . . . 12  |-  f  e. 
_V
19 imaexg 5026 . . . . . . . . . . . 12  |-  ( f  e.  _V  ->  (
f " x )  e.  _V )
2018, 19ax-mp 8 . . . . . . . . . . 11  |-  ( f
" x )  e. 
_V
2120elpw 3631 . . . . . . . . . 10  |-  ( ( f " x )  e.  ~P A  <->  ( f " x )  C_  A )
2217, 21sylibr 203 . . . . . . . . 9  |-  ( ( f  e.  ( J 
Homeo  ~P A )  /\  x  C_  X )  -> 
( f " x
)  e.  ~P A
)
232hmeoopn 17457 . . . . . . . . 9  |-  ( ( f  e.  ( J 
Homeo  ~P A )  /\  x  C_  X )  -> 
( x  e.  J  <->  ( f " x )  e.  ~P A ) )
2422, 23mpbird 223 . . . . . . . 8  |-  ( ( f  e.  ( J 
Homeo  ~P A )  /\  x  C_  X )  ->  x  e.  J )
2524ex 423 . . . . . . 7  |-  ( f  e.  ( J  Homeo  ~P A )  ->  (
x  C_  X  ->  x  e.  J ) )
268, 25syl5 28 . . . . . 6  |-  ( f  e.  ( J  Homeo  ~P A )  ->  (
x  e.  ~P X  ->  x  e.  J ) )
2726ssrdv 3185 . . . . 5  |-  ( f  e.  ( J  Homeo  ~P A )  ->  ~P X  C_  J )
2827exlimiv 1666 . . . 4  |-  ( E. f  f  e.  ( J  Homeo  ~P A
)  ->  ~P X  C_  J )
297, 28sylbi 187 . . 3  |-  ( ( J  Homeo  ~P A
)  =/=  (/)  ->  ~P X  C_  J )
306, 29sylbi 187 . 2  |-  ( J  ~=  ~P A  ->  ~P X  C_  J )
315, 30eqssd 3196 1  |-  ( J  ~=  ~P A  ->  J  =  ~P X
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684    =/= wne 2446   _Vcvv 2788    C_ wss 3152   (/)c0 3455   ~Pcpw 3625   U.cuni 3827   class class class wbr 4023   ran crn 4690   "cima 4692   -->wf 5251   -1-1-onto->wf1o 5254  (class class class)co 5858    Homeo chmeo 17444    ~= chmph 17445
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-suc 4398  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-1o 6479  df-map 6774  df-top 16636  df-topon 16639  df-cn 16957  df-hmeo 17446  df-hmph 17447
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