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Theorem hmeocls 17475
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 17468 . . . 4  |-  ( F  e.  ( J  Homeo  K )  ->  `' F  e.  ( K  Cn  J
) )
2 hmeoopn.1 . . . . 5  |-  X  = 
U. J
32cncls2i 17015 . . . 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 5153 . . . 4  |-  ( `' `' F " A )  =  ( F " A )
65fveq2i 5544 . . 3  |-  ( ( cls `  K ) `
 ( `' `' F " A ) )  =  ( ( cls `  K ) `  ( F " A ) )
7 imacnvcnv 5153 . . 3  |-  ( `' `' F " ( ( cls `  J ) `
 A ) )  =  ( F "
( ( cls `  J
) `  A )
)
84, 6, 73sstr3g 3231 . 2  |-  ( ( F  e.  ( J 
Homeo  K )  /\  A  C_  X )  ->  (
( cls `  K
) `  ( F " A ) )  C_  ( F " ( ( cls `  J ) `
 A ) ) )
9 hmeocn 17467 . . 3  |-  ( F  e.  ( J  Homeo  K )  ->  F  e.  ( J  Cn  K
) )
102cnclsi 17017 . . 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 3209 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 1632    e. wcel 1696    C_ wss 3165   U.cuni 3843   `'ccnv 4704   "cima 4708   ` cfv 5271  (class class class)co 5874   clsccl 16771    Cn ccn 16970    Homeo chmeo 17460
This theorem is referenced by:  reghmph  17500  nrmhmph  17501  snclseqg  17814
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-map 6790  df-top 16652  df-topon 16655  df-cld 16772  df-cls 16774  df-cn 16973  df-hmeo 17462
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