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Theorem hmeoopn 20559
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 20558 . . . 4  |-  ( ( F  e.  ( J
Homeo K )  /\  A  e.  J )  ->  ( F " A )  e.  K )
21ex 432 . . 3  |-  ( F  e.  ( J Homeo K )  ->  ( A  e.  J  ->  ( F
" A )  e.  K ) )
32adantr 463 . 2  |-  ( ( F  e.  ( J
Homeo K )  /\  A  C_  X )  ->  ( A  e.  J  ->  ( F " A )  e.  K ) )
4 hmeocn 20553 . . . . 5  |-  ( F  e.  ( J Homeo K )  ->  F  e.  ( J  Cn  K
) )
5 cnima 20059 . . . . . 6  |-  ( ( F  e.  ( J  Cn  K )  /\  ( F " A )  e.  K )  -> 
( `' F "
( F " A
) )  e.  J
)
65ex 432 . . . . 5  |-  ( F  e.  ( J  Cn  K )  ->  (
( F " A
)  e.  K  -> 
( `' F "
( F " A
) )  e.  J
) )
74, 6syl 17 . . . 4  |-  ( F  e.  ( J Homeo K )  ->  ( ( F " A )  e.  K  ->  ( `' F " ( F " A ) )  e.  J ) )
87adantr 463 . . 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 2402 . . . . . . 7  |-  U. K  =  U. K
119, 10hmeof1o 20557 . . . . . 6  |-  ( F  e.  ( J Homeo K )  ->  F : X
-1-1-onto-> U. K )
12 f1of1 5798 . . . . . 6  |-  ( F : X -1-1-onto-> U. K  ->  F : X -1-1-> U. K )
1311, 12syl 17 . . . . 5  |-  ( F  e.  ( J Homeo K )  ->  F : X -1-1-> U. K )
14 f1imacnv 5815 . . . . 5  |-  ( ( F : X -1-1-> U. K  /\  A  C_  X
)  ->  ( `' F " ( F " A ) )  =  A )
1513, 14sylan 469 . . . 4  |-  ( ( F  e.  ( J
Homeo K )  /\  A  C_  X )  ->  ( `' F " ( F
" A ) )  =  A )
1615eleq1d 2471 . . 3  |-  ( ( F  e.  ( J
Homeo K )  /\  A  C_  X )  ->  (
( `' F "
( F " A
) )  e.  J  <->  A  e.  J ) )
178, 16sylibd 214 . 2  |-  ( ( F  e.  ( J
Homeo K )  /\  A  C_  X )  ->  (
( F " A
)  e.  K  ->  A  e.  J )
)
183, 17impbid 190 1  |-  ( ( F  e.  ( J
Homeo K )  /\  A  C_  X )  ->  ( A  e.  J  <->  ( F " A )  e.  K
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1405    e. wcel 1842    C_ wss 3414   U.cuni 4191   `'ccnv 4822   "cima 4826   -1-1->wf1 5566   -1-1-onto->wf1o 5568  (class class class)co 6278    Cn ccn 20018   Homeochmeo 20546
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-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-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-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-map 7459  df-top 19691  df-topon 19694  df-cn 20021  df-hmeo 20548
This theorem is referenced by:  hmphdis  20589
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