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Theorem hmeocld 20394
Description: Homeomorphisms preserve closedness. (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
hmeocld  |-  ( ( F  e.  ( J
Homeo K )  /\  A  C_  X )  ->  ( A  e.  ( Clsd `  J )  <->  ( F " A )  e.  (
Clsd `  K )
) )

Proof of Theorem hmeocld
StepHypRef Expression
1 hmeocnvcn 20388 . . . 4  |-  ( F  e.  ( J Homeo K )  ->  `' F  e.  ( K  Cn  J
) )
21adantr 465 . . 3  |-  ( ( F  e.  ( J
Homeo K )  /\  A  C_  X )  ->  `' F  e.  ( K  Cn  J ) )
3 imacnvcnv 5478 . . . . 5  |-  ( `' `' F " A )  =  ( F " A )
4 cnclima 19896 . . . . 5  |-  ( ( `' F  e.  ( K  Cn  J )  /\  A  e.  ( Clsd `  J ) )  -> 
( `' `' F " A )  e.  (
Clsd `  K )
)
53, 4syl5eqelr 2550 . . . 4  |-  ( ( `' F  e.  ( K  Cn  J )  /\  A  e.  ( Clsd `  J ) )  -> 
( F " A
)  e.  ( Clsd `  K ) )
65ex 434 . . 3  |-  ( `' F  e.  ( K  Cn  J )  -> 
( A  e.  (
Clsd `  J )  ->  ( F " A
)  e.  ( Clsd `  K ) ) )
72, 6syl 16 . 2  |-  ( ( F  e.  ( J
Homeo K )  /\  A  C_  X )  ->  ( A  e.  ( Clsd `  J )  ->  ( F " A )  e.  ( Clsd `  K
) ) )
8 hmeocn 20387 . . . . 5  |-  ( F  e.  ( J Homeo K )  ->  F  e.  ( J  Cn  K
) )
98adantr 465 . . . 4  |-  ( ( F  e.  ( J
Homeo K )  /\  A  C_  X )  ->  F  e.  ( J  Cn  K
) )
10 cnclima 19896 . . . . 5  |-  ( ( F  e.  ( J  Cn  K )  /\  ( F " A )  e.  ( Clsd `  K
) )  ->  ( `' F " ( F
" A ) )  e.  ( Clsd `  J
) )
1110ex 434 . . . 4  |-  ( F  e.  ( J  Cn  K )  ->  (
( F " A
)  e.  ( Clsd `  K )  ->  ( `' F " ( F
" A ) )  e.  ( Clsd `  J
) ) )
129, 11syl 16 . . 3  |-  ( ( F  e.  ( J
Homeo K )  /\  A  C_  X )  ->  (
( F " A
)  e.  ( Clsd `  K )  ->  ( `' F " ( F
" A ) )  e.  ( Clsd `  J
) ) )
13 hmeoopn.1 . . . . . . 7  |-  X  = 
U. J
14 eqid 2457 . . . . . . 7  |-  U. K  =  U. K
1513, 14hmeof1o 20391 . . . . . 6  |-  ( F  e.  ( J Homeo K )  ->  F : X
-1-1-onto-> U. K )
16 f1of1 5821 . . . . . 6  |-  ( F : X -1-1-onto-> U. K  ->  F : X -1-1-> U. K )
1715, 16syl 16 . . . . 5  |-  ( F  e.  ( J Homeo K )  ->  F : X -1-1-> U. K )
18 f1imacnv 5838 . . . . 5  |-  ( ( F : X -1-1-> U. K  /\  A  C_  X
)  ->  ( `' F " ( F " A ) )  =  A )
1917, 18sylan 471 . . . 4  |-  ( ( F  e.  ( J
Homeo K )  /\  A  C_  X )  ->  ( `' F " ( F
" A ) )  =  A )
2019eleq1d 2526 . . 3  |-  ( ( F  e.  ( J
Homeo K )  /\  A  C_  X )  ->  (
( `' F "
( F " A
) )  e.  (
Clsd `  J )  <->  A  e.  ( Clsd `  J
) ) )
2112, 20sylibd 214 . 2  |-  ( ( F  e.  ( J
Homeo K )  /\  A  C_  X )  ->  (
( F " A
)  e.  ( Clsd `  K )  ->  A  e.  ( Clsd `  J
) ) )
227, 21impbid 191 1  |-  ( ( F  e.  ( J
Homeo K )  /\  A  C_  X )  ->  ( A  e.  ( Clsd `  J )  <->  ( F " A )  e.  (
Clsd `  K )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1395    e. wcel 1819    C_ wss 3471   U.cuni 4251   `'ccnv 5007   "cima 5011   -1-1->wf1 5591   -1-1-onto->wf1o 5593   ` cfv 5594  (class class class)co 6296   Clsdccld 19644    Cn ccn 19852   Homeochmeo 20380
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-map 7440  df-top 19526  df-topon 19529  df-cld 19647  df-cn 19855  df-hmeo 20382
This theorem is referenced by:  cldsubg  20735  reheibor  30519
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