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Theorem hmeocls 17800
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 17793 . . . 4  |-  ( F  e.  ( J  Homeo  K )  ->  `' F  e.  ( K  Cn  J
) )
2 hmeoopn.1 . . . . 5  |-  X  = 
U. J
32cncls2i 17334 . . . 4  |-  ( ( `' F  e.  ( K  Cn  J )  /\  A  C_  X )  -> 
( ( cls `  K
) `  ( `' `' F " A ) )  C_  ( `' `' F " ( ( cls `  J ) `
 A ) ) )
41, 3sylan 458 . . 3  |-  ( ( F  e.  ( J 
Homeo  K )  /\  A  C_  X )  ->  (
( cls `  K
) `  ( `' `' F " A ) )  C_  ( `' `' F " ( ( cls `  J ) `
 A ) ) )
5 imacnvcnv 5334 . . . 4  |-  ( `' `' F " A )  =  ( F " A )
65fveq2i 5731 . . 3  |-  ( ( cls `  K ) `
 ( `' `' F " A ) )  =  ( ( cls `  K ) `  ( F " A ) )
7 imacnvcnv 5334 . . 3  |-  ( `' `' F " ( ( cls `  J ) `
 A ) )  =  ( F "
( ( cls `  J
) `  A )
)
84, 6, 73sstr3g 3388 . 2  |-  ( ( F  e.  ( J 
Homeo  K )  /\  A  C_  X )  ->  (
( cls `  K
) `  ( F " A ) )  C_  ( F " ( ( cls `  J ) `
 A ) ) )
9 hmeocn 17792 . . 3  |-  ( F  e.  ( J  Homeo  K )  ->  F  e.  ( J  Cn  K
) )
102cnclsi 17336 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  A  C_  X )  -> 
( F " (
( cls `  J
) `  A )
)  C_  ( ( cls `  K ) `  ( F " A ) ) )
119, 10sylan 458 . 2  |-  ( ( F  e.  ( J 
Homeo  K )  /\  A  C_  X )  ->  ( F " ( ( cls `  J ) `  A
) )  C_  (
( cls `  K
) `  ( F " A ) ) )
128, 11eqssd 3365 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 359    = wceq 1652    e. wcel 1725    C_ wss 3320   U.cuni 4015   `'ccnv 4877   "cima 4881   ` cfv 5454  (class class class)co 6081   clsccl 17082    Cn ccn 17288    Homeo chmeo 17785
This theorem is referenced by:  reghmph  17825  nrmhmph  17826  snclseqg  18145
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-iin 4096  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-map 7020  df-top 16963  df-topon 16966  df-cld 17083  df-cls 17085  df-cn 17291  df-hmeo 17787
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