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Theorem hmphdis 17820
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 4387 . . . 4  |-  J  C_  ~P U. J
2 hmphdis.1 . . . . 5  |-  X  = 
U. J
32pweqi 3795 . . . 4  |-  ~P X  =  ~P U. J
41, 3sseqtr4i 3373 . . 3  |-  J  C_  ~P X
54a1i 11 . 2  |-  ( J  ~=  ~P A  ->  J  C_  ~P X )
6 hmph 17800 . . 3  |-  ( J  ~=  ~P A  <->  ( J  Homeo  ~P A )  =/=  (/) )
7 n0 3629 . . . 4  |-  ( ( J  Homeo  ~P A
)  =/=  (/)  <->  E. f 
f  e.  ( J 
Homeo  ~P A ) )
8 elpwi 3799 . . . . . . 7  |-  ( x  e.  ~P X  ->  x  C_  X )
9 imassrn 5208 . . . . . . . . . . 11  |-  ( f
" x )  C_  ran  f
10 unipw 4406 . . . . . . . . . . . . . . 15  |-  U. ~P A  =  A
1110eqcomi 2439 . . . . . . . . . . . . . 14  |-  A  = 
U. ~P A
122, 11hmeof1o 17788 . . . . . . . . . . . . 13  |-  ( f  e.  ( J  Homeo  ~P A )  ->  f : X -1-1-onto-> A )
13 f1of 5666 . . . . . . . . . . . . 13  |-  ( f : X -1-1-onto-> A  ->  f : X
--> A )
14 frn 5589 . . . . . . . . . . . . 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 3351 . . . . . . . . . 10  |-  ( ( f  e.  ( J 
Homeo  ~P A )  /\  x  C_  X )  -> 
( f " x
)  C_  A )
18 vex 2951 . . . . . . . . . . . 12  |-  f  e. 
_V
19 imaexg 5209 . . . . . . . . . . . 12  |-  ( f  e.  _V  ->  (
f " x )  e.  _V )
2018, 19ax-mp 8 . . . . . . . . . . 11  |-  ( f
" x )  e. 
_V
2120elpw 3797 . . . . . . . . . 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 17790 . . . . . . . . 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 3346 . . . . 5  |-  ( f  e.  ( J  Homeo  ~P A )  ->  ~P X  C_  J )
2827exlimiv 1644 . . . 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 3357 1  |-  ( J  ~=  ~P A  ->  J  =  ~P X
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   E.wex 1550    = wceq 1652    e. wcel 1725    =/= wne 2598   _Vcvv 2948    C_ wss 3312   (/)c0 3620   ~Pcpw 3791   U.cuni 4007   class class class wbr 4204   ran crn 4871   "cima 4873   -->wf 5442   -1-1-onto->wf1o 5445  (class class class)co 6073    Homeo chmeo 17777    ~= chmph 17778
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-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-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-top 16955  df-topon 16958  df-cn 17283  df-hmeo 17779  df-hmph 17780
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