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Theorem hmeocld 12491
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 12485 . . . 4  |-  ( F  e.  ( J Homeo K )  ->  `' F  e.  ( K  Cn  J
) )
21adantr 274 . . 3  |-  ( ( F  e.  ( J
Homeo K )  /\  A  C_  X )  ->  `' F  e.  ( K  Cn  J ) )
3 imacnvcnv 5003 . . . . 5  |-  ( `' `' F " A )  =  ( F " A )
4 cnclima 12402 . . . . 5  |-  ( ( `' F  e.  ( K  Cn  J )  /\  A  e.  ( Clsd `  J ) )  -> 
( `' `' F " A )  e.  (
Clsd `  K )
)
53, 4eqeltrrid 2227 . . . 4  |-  ( ( `' F  e.  ( K  Cn  J )  /\  A  e.  ( Clsd `  J ) )  -> 
( F " A
)  e.  ( Clsd `  K ) )
65ex 114 . . 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 12484 . . . . 5  |-  ( F  e.  ( J Homeo K )  ->  F  e.  ( J  Cn  K
) )
98adantr 274 . . . 4  |-  ( ( F  e.  ( J
Homeo K )  /\  A  C_  X )  ->  F  e.  ( J  Cn  K
) )
10 cnclima 12402 . . . . 5  |-  ( ( F  e.  ( J  Cn  K )  /\  ( F " A )  e.  ( Clsd `  K
) )  ->  ( `' F " ( F
" A ) )  e.  ( Clsd `  J
) )
1110ex 114 . . . 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 2139 . . . . . . 7  |-  U. K  =  U. K
1513, 14hmeof1o 12488 . . . . . 6  |-  ( F  e.  ( J Homeo K )  ->  F : X
-1-1-onto-> U. K )
16 f1of1 5366 . . . . . 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 5384 . . . . 5  |-  ( ( F : X -1-1-> U. K  /\  A  C_  X
)  ->  ( `' F " ( F " A ) )  =  A )
1917, 18sylan 281 . . . 4  |-  ( ( F  e.  ( J
Homeo K )  /\  A  C_  X )  ->  ( `' F " ( F
" A ) )  =  A )
2019eleq1d 2208 . . 3  |-  ( ( F  e.  ( J
Homeo K )  /\  A  C_  X )  ->  (
( `' F "
( F " A
) )  e.  (
Clsd `  J )  <->  A  e.  ( Clsd `  J
) ) )
2112, 20sylibd 148 . 2  |-  ( ( F  e.  ( J
Homeo K )  /\  A  C_  X )  ->  (
( F " A
)  e.  ( Clsd `  K )  ->  A  e.  ( Clsd `  J
) ) )
227, 21impbid 128 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 103    <-> wb 104    = wceq 1331    e. wcel 1480    C_ wss 3071   U.cuni 3736   `'ccnv 4538   "cima 4542   -1-1->wf1 5120   -1-1-onto->wf1o 5122   ` cfv 5123  (class class class)co 5774   Clsdccld 12271    Cn ccn 12364   Homeochmeo 12479
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-map 6544  df-top 12175  df-topon 12188  df-cld 12274  df-cn 12367  df-hmeo 12480
This theorem is referenced by: (None)
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