MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  hmeoopn Unicode version

Theorem hmeoopn 17457
Description: Homeomorphisms preserve openness. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 25-Aug-2015.)
Hypothesis
Ref Expression
hmeoopn.1  |-  X  = 
U. J
Assertion
Ref Expression
hmeoopn  |-  ( ( F  e.  ( J 
Homeo  K )  /\  A  C_  X )  ->  ( A  e.  J  <->  ( F " A )  e.  K
) )

Proof of Theorem hmeoopn
StepHypRef Expression
1 hmeoima 17456 . . . 4  |-  ( ( F  e.  ( J 
Homeo  K )  /\  A  e.  J )  ->  ( F " A )  e.  K )
21ex 423 . . 3  |-  ( F  e.  ( J  Homeo  K )  ->  ( A  e.  J  ->  ( F
" A )  e.  K ) )
32adantr 451 . 2  |-  ( ( F  e.  ( J 
Homeo  K )  /\  A  C_  X )  ->  ( A  e.  J  ->  ( F " A )  e.  K ) )
4 hmeocn 17451 . . . . 5  |-  ( F  e.  ( J  Homeo  K )  ->  F  e.  ( J  Cn  K
) )
5 cnima 16994 . . . . . 6  |-  ( ( F  e.  ( J  Cn  K )  /\  ( F " A )  e.  K )  -> 
( `' F "
( F " A
) )  e.  J
)
65ex 423 . . . . 5  |-  ( F  e.  ( J  Cn  K )  ->  (
( F " A
)  e.  K  -> 
( `' F "
( F " A
) )  e.  J
) )
74, 6syl 15 . . . 4  |-  ( F  e.  ( J  Homeo  K )  ->  ( ( F " A )  e.  K  ->  ( `' F " ( F " A ) )  e.  J ) )
87adantr 451 . . 3  |-  ( ( F  e.  ( J 
Homeo  K )  /\  A  C_  X )  ->  (
( F " A
)  e.  K  -> 
( `' F "
( F " A
) )  e.  J
) )
9 hmeoopn.1 . . . . . . 7  |-  X  = 
U. J
10 eqid 2283 . . . . . . 7  |-  U. K  =  U. K
119, 10hmeof1o 17455 . . . . . 6  |-  ( F  e.  ( J  Homeo  K )  ->  F : X
-1-1-onto-> U. K )
12 f1of1 5471 . . . . . 6  |-  ( F : X -1-1-onto-> U. K  ->  F : X -1-1-> U. K )
1311, 12syl 15 . . . . 5  |-  ( F  e.  ( J  Homeo  K )  ->  F : X -1-1-> U. K )
14 f1imacnv 5489 . . . . 5  |-  ( ( F : X -1-1-> U. K  /\  A  C_  X
)  ->  ( `' F " ( F " A ) )  =  A )
1513, 14sylan 457 . . . 4  |-  ( ( F  e.  ( J 
Homeo  K )  /\  A  C_  X )  ->  ( `' F " ( F
" A ) )  =  A )
1615eleq1d 2349 . . 3  |-  ( ( F  e.  ( J 
Homeo  K )  /\  A  C_  X )  ->  (
( `' F "
( F " A
) )  e.  J  <->  A  e.  J ) )
178, 16sylibd 205 . 2  |-  ( ( F  e.  ( J 
Homeo  K )  /\  A  C_  X )  ->  (
( F " A
)  e.  K  ->  A  e.  J )
)
183, 17impbid 183 1  |-  ( ( F  e.  ( J 
Homeo  K )  /\  A  C_  X )  ->  ( A  e.  J  <->  ( F " A )  e.  K
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    C_ wss 3152   U.cuni 3827   `'ccnv 4688   "cima 4692   -1-1->wf1 5252   -1-1-onto->wf1o 5254  (class class class)co 5858    Cn ccn 16954    Homeo chmeo 17444
This theorem is referenced by:  hmphdis  17487
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-map 6774  df-top 16636  df-topon 16639  df-cn 16957  df-hmeo 17446
  Copyright terms: Public domain W3C validator