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Theorem hmphdis 20025
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 4671 . . . 4  |-  J  C_  ~P U. J
2 hmphdis.1 . . . . 5  |-  X  = 
U. J
32pweqi 4007 . . . 4  |-  ~P X  =  ~P U. J
41, 3sseqtr4i 3530 . . 3  |-  J  C_  ~P X
54a1i 11 . 2  |-  ( J  ~=  ~P A  ->  J  C_  ~P X )
6 hmph 20005 . . 3  |-  ( J  ~=  ~P A  <->  ( J Homeo ~P A )  =/=  (/) )
7 n0 3787 . . . 4  |-  ( ( J Homeo ~P A )  =/=  (/)  <->  E. f  f  e.  ( J Homeo ~P A
) )
8 elpwi 4012 . . . . . . 7  |-  ( x  e.  ~P X  ->  x  C_  X )
9 imassrn 5339 . . . . . . . . . . 11  |-  ( f
" x )  C_  ran  f
10 unipw 4690 . . . . . . . . . . . . . . 15  |-  U. ~P A  =  A
1110eqcomi 2473 . . . . . . . . . . . . . 14  |-  A  = 
U. ~P A
122, 11hmeof1o 19993 . . . . . . . . . . . . 13  |-  ( f  e.  ( J Homeo ~P A )  ->  f : X -1-1-onto-> A )
13 f1of 5807 . . . . . . . . . . . . 13  |-  ( f : X -1-1-onto-> A  ->  f : X
--> A )
14 frn 5728 . . . . . . . . . . . . 13  |-  ( f : X --> A  ->  ran  f  C_  A )
1512, 13, 143syl 20 . . . . . . . . . . . 12  |-  ( f  e.  ( J Homeo ~P A )  ->  ran  f  C_  A )
1615adantr 465 . . . . . . . . . . 11  |-  ( ( f  e.  ( J
Homeo ~P A )  /\  x  C_  X )  ->  ran  f  C_  A )
179, 16syl5ss 3508 . . . . . . . . . 10  |-  ( ( f  e.  ( J
Homeo ~P A )  /\  x  C_  X )  -> 
( f " x
)  C_  A )
18 vex 3109 . . . . . . . . . . . 12  |-  f  e. 
_V
19 imaexg 6711 . . . . . . . . . . . 12  |-  ( f  e.  _V  ->  (
f " x )  e.  _V )
2018, 19ax-mp 5 . . . . . . . . . . 11  |-  ( f
" x )  e. 
_V
2120elpw 4009 . . . . . . . . . 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 19995 . . . . . . . . 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 434 . . . . . . 7  |-  ( f  e.  ( J Homeo ~P A )  ->  (
x  C_  X  ->  x  e.  J ) )
268, 25syl5 32 . . . . . 6  |-  ( f  e.  ( J Homeo ~P A )  ->  (
x  e.  ~P X  ->  x  e.  J ) )
2726ssrdv 3503 . . . . 5  |-  ( f  e.  ( J Homeo ~P A )  ->  ~P X  C_  J )
2827exlimiv 1693 . . . 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 3514 1  |-  ( J  ~=  ~P A  ->  J  =  ~P X
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1374   E.wex 1591    e. wcel 1762    =/= wne 2655   _Vcvv 3106    C_ wss 3469   (/)c0 3778   ~Pcpw 4003   U.cuni 4238   class class class wbr 4440   ran crn 4993   "cima 4995   -->wf 5575   -1-1-onto->wf1o 5578  (class class class)co 6275   Homeochmeo 19982    ~= chmph 19983
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-1st 6774  df-2nd 6775  df-1o 7120  df-map 7412  df-top 19159  df-topon 19162  df-cn 19487  df-hmeo 19984  df-hmph 19985
This theorem is referenced by: (None)
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