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Theorem hmphdis 19391
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 4544 . . . 4  |-  J  C_  ~P U. J
2 hmphdis.1 . . . . 5  |-  X  = 
U. J
32pweqi 3885 . . . 4  |-  ~P X  =  ~P U. J
41, 3sseqtr4i 3410 . . 3  |-  J  C_  ~P X
54a1i 11 . 2  |-  ( J  ~=  ~P A  ->  J  C_  ~P X )
6 hmph 19371 . . 3  |-  ( J  ~=  ~P A  <->  ( J Homeo ~P A )  =/=  (/) )
7 n0 3667 . . . 4  |-  ( ( J Homeo ~P A )  =/=  (/)  <->  E. f  f  e.  ( J Homeo ~P A
) )
8 elpwi 3890 . . . . . . 7  |-  ( x  e.  ~P X  ->  x  C_  X )
9 imassrn 5201 . . . . . . . . . . 11  |-  ( f
" x )  C_  ran  f
10 unipw 4563 . . . . . . . . . . . . . . 15  |-  U. ~P A  =  A
1110eqcomi 2447 . . . . . . . . . . . . . 14  |-  A  = 
U. ~P A
122, 11hmeof1o 19359 . . . . . . . . . . . . 13  |-  ( f  e.  ( J Homeo ~P A )  ->  f : X -1-1-onto-> A )
13 f1of 5662 . . . . . . . . . . . . 13  |-  ( f : X -1-1-onto-> A  ->  f : X
--> A )
14 frn 5586 . . . . . . . . . . . . 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 3388 . . . . . . . . . 10  |-  ( ( f  e.  ( J
Homeo ~P A )  /\  x  C_  X )  -> 
( f " x
)  C_  A )
18 vex 2996 . . . . . . . . . . . 12  |-  f  e. 
_V
19 imaexg 6536 . . . . . . . . . . . 12  |-  ( f  e.  _V  ->  (
f " x )  e.  _V )
2018, 19ax-mp 5 . . . . . . . . . . 11  |-  ( f
" x )  e. 
_V
2120elpw 3887 . . . . . . . . . 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 19361 . . . . . . . . 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 3383 . . . . 5  |-  ( f  e.  ( J Homeo ~P A )  ->  ~P X  C_  J )
2827exlimiv 1688 . . . 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 3394 1  |-  ( J  ~=  ~P A  ->  J  =  ~P X
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369   E.wex 1586    e. wcel 1756    =/= wne 2620   _Vcvv 2993    C_ wss 3349   (/)c0 3658   ~Pcpw 3881   U.cuni 4112   class class class wbr 4313   ran crn 4862   "cima 4864   -->wf 5435   -1-1-onto->wf1o 5438  (class class class)co 6112   Homeochmeo 19348    ~= chmph 19349
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-id 4657  df-suc 4746  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-1st 6598  df-2nd 6599  df-1o 6941  df-map 7237  df-top 18525  df-topon 18528  df-cn 18853  df-hmeo 19350  df-hmph 19351
This theorem is referenced by: (None)
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