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Theorem hmeocld 19338
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 19332 . . . 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 5301 . . . . 5  |-  ( `' `' F " A )  =  ( F " A )
4 cnclima 18870 . . . . 5  |-  ( ( `' F  e.  ( K  Cn  J )  /\  A  e.  ( Clsd `  J ) )  -> 
( `' `' F " A )  e.  (
Clsd `  K )
)
53, 4syl5eqelr 2526 . . . 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 19331 . . . . 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 18870 . . . . 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 2441 . . . . . . 7  |-  U. K  =  U. K
1513, 14hmeof1o 19335 . . . . . 6  |-  ( F  e.  ( J Homeo K )  ->  F : X
-1-1-onto-> U. K )
16 f1of1 5638 . . . . . 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 5655 . . . . 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 2507 . . 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 1369    e. wcel 1756    C_ wss 3326   U.cuni 4089   `'ccnv 4837   "cima 4841   -1-1->wf1 5413   -1-1-onto->wf1o 5415   ` cfv 5416  (class class class)co 6089   Clsdccld 18618    Cn ccn 18826   Homeochmeo 19324
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3185  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-br 4291  df-opab 4349  df-mpt 4350  df-id 4634  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-map 7214  df-top 18501  df-topon 18504  df-cld 18621  df-cn 18829  df-hmeo 19326
This theorem is referenced by:  cldsubg  19679  reheibor  28735
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