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

Theorem hmeoopn 19997
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 19996 . . . 4  |-  ( ( F  e.  ( J
Homeo K )  /\  A  e.  J )  ->  ( F " A )  e.  K )
21ex 434 . . 3  |-  ( F  e.  ( J Homeo K )  ->  ( A  e.  J  ->  ( F
" A )  e.  K ) )
32adantr 465 . 2  |-  ( ( F  e.  ( J
Homeo K )  /\  A  C_  X )  ->  ( A  e.  J  ->  ( F " A )  e.  K ) )
4 hmeocn 19991 . . . . 5  |-  ( F  e.  ( J Homeo K )  ->  F  e.  ( J  Cn  K
) )
5 cnima 19527 . . . . . 6  |-  ( ( F  e.  ( J  Cn  K )  /\  ( F " A )  e.  K )  -> 
( `' F "
( F " A
) )  e.  J
)
65ex 434 . . . . 5  |-  ( F  e.  ( J  Cn  K )  ->  (
( F " A
)  e.  K  -> 
( `' F "
( F " A
) )  e.  J
) )
74, 6syl 16 . . . 4  |-  ( F  e.  ( J Homeo K )  ->  ( ( F " A )  e.  K  ->  ( `' F " ( F " A ) )  e.  J ) )
87adantr 465 . . 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 2462 . . . . . . 7  |-  U. K  =  U. K
119, 10hmeof1o 19995 . . . . . 6  |-  ( F  e.  ( J Homeo K )  ->  F : X
-1-1-onto-> U. K )
12 f1of1 5808 . . . . . 6  |-  ( F : X -1-1-onto-> U. K  ->  F : X -1-1-> U. K )
1311, 12syl 16 . . . . 5  |-  ( F  e.  ( J Homeo K )  ->  F : X -1-1-> U. K )
14 f1imacnv 5825 . . . . 5  |-  ( ( F : X -1-1-> U. K  /\  A  C_  X
)  ->  ( `' F " ( F " A ) )  =  A )
1513, 14sylan 471 . . . 4  |-  ( ( F  e.  ( J
Homeo K )  /\  A  C_  X )  ->  ( `' F " ( F
" A ) )  =  A )
1615eleq1d 2531 . . 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 191 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 369    = wceq 1374    e. wcel 1762    C_ wss 3471   U.cuni 4240   `'ccnv 4993   "cima 4997   -1-1->wf1 5578   -1-1-onto->wf1o 5580  (class class class)co 6277    Cn ccn 19486   Homeochmeo 19984
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 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569
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 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-sbc 3327  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-map 7414  df-top 19161  df-topon 19164  df-cn 19489  df-hmeo 19986
This theorem is referenced by:  hmphdis  20027
  Copyright terms: Public domain W3C validator