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Theorem hmphdis 20589
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 4622 . . . 4  |-  J  C_  ~P U. J
2 hmphdis.1 . . . . 5  |-  X  = 
U. J
32pweqi 3959 . . . 4  |-  ~P X  =  ~P U. J
41, 3sseqtr4i 3475 . . 3  |-  J  C_  ~P X
54a1i 11 . 2  |-  ( J  ~=  ~P A  ->  J  C_  ~P X )
6 hmph 20569 . . 3  |-  ( J  ~=  ~P A  <->  ( J Homeo ~P A )  =/=  (/) )
7 n0 3748 . . . 4  |-  ( ( J Homeo ~P A )  =/=  (/)  <->  E. f  f  e.  ( J Homeo ~P A
) )
8 elpwi 3964 . . . . . . 7  |-  ( x  e.  ~P X  ->  x  C_  X )
9 imassrn 5168 . . . . . . . . . . 11  |-  ( f
" x )  C_  ran  f
10 unipw 4641 . . . . . . . . . . . . . . 15  |-  U. ~P A  =  A
1110eqcomi 2415 . . . . . . . . . . . . . 14  |-  A  = 
U. ~P A
122, 11hmeof1o 20557 . . . . . . . . . . . . 13  |-  ( f  e.  ( J Homeo ~P A )  ->  f : X -1-1-onto-> A )
13 f1of 5799 . . . . . . . . . . . . 13  |-  ( f : X -1-1-onto-> A  ->  f : X
--> A )
14 frn 5720 . . . . . . . . . . . . 13  |-  ( f : X --> A  ->  ran  f  C_  A )
1512, 13, 143syl 18 . . . . . . . . . . . 12  |-  ( f  e.  ( J Homeo ~P A )  ->  ran  f  C_  A )
1615adantr 463 . . . . . . . . . . 11  |-  ( ( f  e.  ( J
Homeo ~P A )  /\  x  C_  X )  ->  ran  f  C_  A )
179, 16syl5ss 3453 . . . . . . . . . 10  |-  ( ( f  e.  ( J
Homeo ~P A )  /\  x  C_  X )  -> 
( f " x
)  C_  A )
18 vex 3062 . . . . . . . . . . . 12  |-  f  e. 
_V
19 imaexg 6721 . . . . . . . . . . . 12  |-  ( f  e.  _V  ->  (
f " x )  e.  _V )
2018, 19ax-mp 5 . . . . . . . . . . 11  |-  ( f
" x )  e. 
_V
2120elpw 3961 . . . . . . . . . 10  |-  ( ( f " x )  e.  ~P A  <->  ( f " x )  C_  A )
2217, 21sylibr 212 . . . . . . . . 9  |-  ( ( f  e.  ( J
Homeo ~P A )  /\  x  C_  X )  -> 
( f " x
)  e.  ~P A
)
232hmeoopn 20559 . . . . . . . . 9  |-  ( ( f  e.  ( J
Homeo ~P A )  /\  x  C_  X )  -> 
( x  e.  J  <->  ( f " x )  e.  ~P A ) )
2422, 23mpbird 232 . . . . . . . 8  |-  ( ( f  e.  ( J
Homeo ~P A )  /\  x  C_  X )  ->  x  e.  J )
2524ex 432 . . . . . . 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 3448 . . . . 5  |-  ( f  e.  ( J Homeo ~P A )  ->  ~P X  C_  J )
2827exlimiv 1743 . . . 4  |-  ( E. f  f  e.  ( J Homeo ~P A )  ->  ~P X  C_  J )
297, 28sylbi 195 . . 3  |-  ( ( J Homeo ~P A )  =/=  (/)  ->  ~P X  C_  J )
306, 29sylbi 195 . 2  |-  ( J  ~=  ~P A  ->  ~P X  C_  J )
315, 30eqssd 3459 1  |-  ( J  ~=  ~P A  ->  J  =  ~P X
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1405   E.wex 1633    e. wcel 1842    =/= wne 2598   _Vcvv 3059    C_ wss 3414   (/)c0 3738   ~Pcpw 3955   U.cuni 4191   class class class wbr 4395   ran crn 4824   "cima 4826   -->wf 5565   -1-1-onto->wf1o 5568  (class class class)co 6278   Homeochmeo 20546    ~= chmph 20547
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-1st 6784  df-2nd 6785  df-1o 7167  df-map 7459  df-top 19691  df-topon 19694  df-cn 20021  df-hmeo 20548  df-hmph 20549
This theorem is referenced by: (None)
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