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Theorem hmeoopn 17473
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 17472 . . . 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 17467 . . . . 5  |-  ( F  e.  ( J  Homeo  K )  ->  F  e.  ( J  Cn  K
) )
5 cnima 17010 . . . . . 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 2296 . . . . . . 7  |-  U. K  =  U. K
119, 10hmeof1o 17471 . . . . . 6  |-  ( F  e.  ( J  Homeo  K )  ->  F : X
-1-1-onto-> U. K )
12 f1of1 5487 . . . . . 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 5505 . . . . 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 2362 . . 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 1632    e. wcel 1696    C_ wss 3165   U.cuni 3843   `'ccnv 4704   "cima 4708   -1-1->wf1 5268   -1-1-onto->wf1o 5270  (class class class)co 5874    Cn ccn 16970    Homeo chmeo 17460
This theorem is referenced by:  hmphdis  17503
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-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-map 6790  df-top 16652  df-topon 16655  df-cn 16973  df-hmeo 17462
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