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Theorem hmphdis 17503
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 4222 . . . 4  |-  J  C_  ~P U. J
2 hmphdis.1 . . . . 5  |-  X  = 
U. J
32pweqi 3642 . . . 4  |-  ~P X  =  ~P U. J
41, 3sseqtr4i 3224 . . 3  |-  J  C_  ~P X
54a1i 10 . 2  |-  ( J  ~=  ~P A  ->  J  C_  ~P X )
6 hmph 17483 . . 3  |-  ( J  ~=  ~P A  <->  ( J  Homeo  ~P A )  =/=  (/) )
7 n0 3477 . . . 4  |-  ( ( J  Homeo  ~P A
)  =/=  (/)  <->  E. f 
f  e.  ( J 
Homeo  ~P A ) )
8 elpwi 3646 . . . . . . 7  |-  ( x  e.  ~P X  ->  x  C_  X )
9 imassrn 5041 . . . . . . . . . . 11  |-  ( f
" x )  C_  ran  f
10 unipw 4240 . . . . . . . . . . . . . . 15  |-  U. ~P A  =  A
1110eqcomi 2300 . . . . . . . . . . . . . 14  |-  A  = 
U. ~P A
122, 11hmeof1o 17471 . . . . . . . . . . . . 13  |-  ( f  e.  ( J  Homeo  ~P A )  ->  f : X -1-1-onto-> A )
13 f1of 5488 . . . . . . . . . . . . 13  |-  ( f : X -1-1-onto-> A  ->  f : X
--> A )
14 frn 5411 . . . . . . . . . . . . 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 3203 . . . . . . . . . 10  |-  ( ( f  e.  ( J 
Homeo  ~P A )  /\  x  C_  X )  -> 
( f " x
)  C_  A )
18 vex 2804 . . . . . . . . . . . 12  |-  f  e. 
_V
19 imaexg 5042 . . . . . . . . . . . 12  |-  ( f  e.  _V  ->  (
f " x )  e.  _V )
2018, 19ax-mp 8 . . . . . . . . . . 11  |-  ( f
" x )  e. 
_V
2120elpw 3644 . . . . . . . . . 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 17473 . . . . . . . . 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 3198 . . . . 5  |-  ( f  e.  ( J  Homeo  ~P A )  ->  ~P X  C_  J )
2827exlimiv 1624 . . . 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 3209 1  |-  ( J  ~=  ~P A  ->  J  =  ~P X
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   E.wex 1531    = wceq 1632    e. wcel 1696    =/= wne 2459   _Vcvv 2801    C_ wss 3165   (/)c0 3468   ~Pcpw 3638   U.cuni 3843   class class class wbr 4039   ran crn 4706   "cima 4708   -->wf 5267   -1-1-onto->wf1o 5270  (class class class)co 5874    Homeo chmeo 17460    ~= chmph 17461
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-suc 4414  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-1o 6495  df-map 6790  df-top 16652  df-topon 16655  df-cn 16973  df-hmeo 17462  df-hmph 17463
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