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Theorem hmeocls 17459
Description: Homeomorphisms preserve closures. (Contributed by Mario Carneiro, 25-Aug-2015.)
Hypothesis
Ref Expression
hmeoopn.1  |-  X  = 
U. J
Assertion
Ref Expression
hmeocls  |-  ( ( F  e.  ( J 
Homeo  K )  /\  A  C_  X )  ->  (
( cls `  K
) `  ( F " A ) )  =  ( F " (
( cls `  J
) `  A )
) )

Proof of Theorem hmeocls
StepHypRef Expression
1 hmeocnvcn 17452 . . . 4  |-  ( F  e.  ( J  Homeo  K )  ->  `' F  e.  ( K  Cn  J
) )
2 hmeoopn.1 . . . . 5  |-  X  = 
U. J
32cncls2i 16999 . . . 4  |-  ( ( `' F  e.  ( K  Cn  J )  /\  A  C_  X )  -> 
( ( cls `  K
) `  ( `' `' F " A ) )  C_  ( `' `' F " ( ( cls `  J ) `
 A ) ) )
41, 3sylan 457 . . 3  |-  ( ( F  e.  ( J 
Homeo  K )  /\  A  C_  X )  ->  (
( cls `  K
) `  ( `' `' F " A ) )  C_  ( `' `' F " ( ( cls `  J ) `
 A ) ) )
5 imacnvcnv 5137 . . . 4  |-  ( `' `' F " A )  =  ( F " A )
65fveq2i 5528 . . 3  |-  ( ( cls `  K ) `
 ( `' `' F " A ) )  =  ( ( cls `  K ) `  ( F " A ) )
7 imacnvcnv 5137 . . 3  |-  ( `' `' F " ( ( cls `  J ) `
 A ) )  =  ( F "
( ( cls `  J
) `  A )
)
84, 6, 73sstr3g 3218 . 2  |-  ( ( F  e.  ( J 
Homeo  K )  /\  A  C_  X )  ->  (
( cls `  K
) `  ( F " A ) )  C_  ( F " ( ( cls `  J ) `
 A ) ) )
9 hmeocn 17451 . . 3  |-  ( F  e.  ( J  Homeo  K )  ->  F  e.  ( J  Cn  K
) )
102cnclsi 17001 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  A  C_  X )  -> 
( F " (
( cls `  J
) `  A )
)  C_  ( ( cls `  K ) `  ( F " A ) ) )
119, 10sylan 457 . 2  |-  ( ( F  e.  ( J 
Homeo  K )  /\  A  C_  X )  ->  ( F " ( ( cls `  J ) `  A
) )  C_  (
( cls `  K
) `  ( F " A ) ) )
128, 11eqssd 3196 1  |-  ( ( F  e.  ( J 
Homeo  K )  /\  A  C_  X )  ->  (
( cls `  K
) `  ( F " A ) )  =  ( F " (
( cls `  J
) `  A )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    C_ wss 3152   U.cuni 3827   `'ccnv 4688   "cima 4692   ` cfv 5255  (class class class)co 5858   clsccl 16755    Cn ccn 16954    Homeo chmeo 17444
This theorem is referenced by:  reghmph  17484  nrmhmph  17485  snclseqg  17798
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-map 6774  df-top 16636  df-topon 16639  df-cld 16756  df-cls 16758  df-cn 16957  df-hmeo 17446
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