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Theorem hmeocld 19996
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 19990 . . . 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 5463 . . . . 5  |-  ( `' `' F " A )  =  ( F " A )
4 cnclima 19528 . . . . 5  |-  ( ( `' F  e.  ( K  Cn  J )  /\  A  e.  ( Clsd `  J ) )  -> 
( `' `' F " A )  e.  (
Clsd `  K )
)
53, 4syl5eqelr 2553 . . . 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 19989 . . . . 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 19528 . . . . 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 2460 . . . . . . 7  |-  U. K  =  U. K
1513, 14hmeof1o 19993 . . . . . 6  |-  ( F  e.  ( J Homeo K )  ->  F : X
-1-1-onto-> U. K )
16 f1of1 5806 . . . . . 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 5823 . . . . 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 2529 . . 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 1374    e. wcel 1762    C_ wss 3469   U.cuni 4238   `'ccnv 4991   "cima 4995   -1-1->wf1 5576   -1-1-onto->wf1o 5578   ` cfv 5579  (class class class)co 6275   Clsdccld 19276    Cn ccn 19484   Homeochmeo 19982
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 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
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 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-map 7412  df-top 19159  df-topon 19162  df-cld 19279  df-cn 19487  df-hmeo 19984
This theorem is referenced by:  cldsubg  20337  reheibor  29925
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