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Theorem hmeocld 13674
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 13668 . . . 4  |-  ( F  e.  ( J Homeo K )  ->  `' F  e.  ( K  Cn  J
) )
21adantr 276 . . 3  |-  ( ( F  e.  ( J
Homeo K )  /\  A  C_  X )  ->  `' F  e.  ( K  Cn  J ) )
3 imacnvcnv 5091 . . . . 5  |-  ( `' `' F " A )  =  ( F " A )
4 cnclima 13585 . . . . 5  |-  ( ( `' F  e.  ( K  Cn  J )  /\  A  e.  ( Clsd `  J ) )  -> 
( `' `' F " A )  e.  (
Clsd `  K )
)
53, 4eqeltrrid 2265 . . . 4  |-  ( ( `' F  e.  ( K  Cn  J )  /\  A  e.  ( Clsd `  J ) )  -> 
( F " A
)  e.  ( Clsd `  K ) )
65ex 115 . . 3  |-  ( `' F  e.  ( K  Cn  J )  -> 
( A  e.  (
Clsd `  J )  ->  ( F " A
)  e.  ( Clsd `  K ) ) )
72, 6syl 14 . 2  |-  ( ( F  e.  ( J
Homeo K )  /\  A  C_  X )  ->  ( A  e.  ( Clsd `  J )  ->  ( F " A )  e.  ( Clsd `  K
) ) )
8 hmeocn 13667 . . . . 5  |-  ( F  e.  ( J Homeo K )  ->  F  e.  ( J  Cn  K
) )
98adantr 276 . . . 4  |-  ( ( F  e.  ( J
Homeo K )  /\  A  C_  X )  ->  F  e.  ( J  Cn  K
) )
10 cnclima 13585 . . . . 5  |-  ( ( F  e.  ( J  Cn  K )  /\  ( F " A )  e.  ( Clsd `  K
) )  ->  ( `' F " ( F
" A ) )  e.  ( Clsd `  J
) )
1110ex 115 . . . 4  |-  ( F  e.  ( J  Cn  K )  ->  (
( F " A
)  e.  ( Clsd `  K )  ->  ( `' F " ( F
" A ) )  e.  ( Clsd `  J
) ) )
129, 11syl 14 . . 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 2177 . . . . . . 7  |-  U. K  =  U. K
1513, 14hmeof1o 13671 . . . . . 6  |-  ( F  e.  ( J Homeo K )  ->  F : X
-1-1-onto-> U. K )
16 f1of1 5458 . . . . . 6  |-  ( F : X -1-1-onto-> U. K  ->  F : X -1-1-> U. K )
1715, 16syl 14 . . . . 5  |-  ( F  e.  ( J Homeo K )  ->  F : X -1-1-> U. K )
18 f1imacnv 5476 . . . . 5  |-  ( ( F : X -1-1-> U. K  /\  A  C_  X
)  ->  ( `' F " ( F " A ) )  =  A )
1917, 18sylan 283 . . . 4  |-  ( ( F  e.  ( J
Homeo K )  /\  A  C_  X )  ->  ( `' F " ( F
" A ) )  =  A )
2019eleq1d 2246 . . 3  |-  ( ( F  e.  ( J
Homeo K )  /\  A  C_  X )  ->  (
( `' F "
( F " A
) )  e.  (
Clsd `  J )  <->  A  e.  ( Clsd `  J
) ) )
2112, 20sylibd 149 . 2  |-  ( ( F  e.  ( J
Homeo K )  /\  A  C_  X )  ->  (
( F " A
)  e.  ( Clsd `  K )  ->  A  e.  ( Clsd `  J
) ) )
227, 21impbid 129 1  |-  ( ( F  e.  ( J
Homeo K )  /\  A  C_  X )  ->  ( A  e.  ( Clsd `  J )  <->  ( F " A )  e.  (
Clsd `  K )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1353    e. wcel 2148    C_ wss 3129   U.cuni 3809   `'ccnv 4624   "cima 4628   -1-1->wf1 5211   -1-1-onto->wf1o 5213   ` cfv 5214  (class class class)co 5871   Clsdccld 13454    Cn ccn 13547   Homeochmeo 13662
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4120  ax-pow 4173  ax-pr 4208  ax-un 4432  ax-setind 4535
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-iun 3888  df-br 4003  df-opab 4064  df-mpt 4065  df-id 4292  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-res 4637  df-ima 4638  df-iota 5176  df-fun 5216  df-fn 5217  df-f 5218  df-f1 5219  df-fo 5220  df-f1o 5221  df-fv 5222  df-ov 5874  df-oprab 5875  df-mpo 5876  df-1st 6137  df-2nd 6138  df-map 6646  df-top 13358  df-topon 13371  df-cld 13457  df-cn 13550  df-hmeo 13663
This theorem is referenced by: (None)
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