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Theorem hmeocls 19454
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 19447 . . . 4  |-  ( F  e.  ( J Homeo K )  ->  `' F  e.  ( K  Cn  J
) )
2 hmeoopn.1 . . . . 5  |-  X  = 
U. J
32cncls2i 18987 . . . 4  |-  ( ( `' F  e.  ( K  Cn  J )  /\  A  C_  X )  -> 
( ( cls `  K
) `  ( `' `' F " A ) )  C_  ( `' `' F " ( ( cls `  J ) `
 A ) ) )
41, 3sylan 471 . . 3  |-  ( ( F  e.  ( J
Homeo K )  /\  A  C_  X )  ->  (
( cls `  K
) `  ( `' `' F " A ) )  C_  ( `' `' F " ( ( cls `  J ) `
 A ) ) )
5 imacnvcnv 5398 . . . 4  |-  ( `' `' F " A )  =  ( F " A )
65fveq2i 5789 . . 3  |-  ( ( cls `  K ) `
 ( `' `' F " A ) )  =  ( ( cls `  K ) `  ( F " A ) )
7 imacnvcnv 5398 . . 3  |-  ( `' `' F " ( ( cls `  J ) `
 A ) )  =  ( F "
( ( cls `  J
) `  A )
)
84, 6, 73sstr3g 3491 . 2  |-  ( ( F  e.  ( J
Homeo K )  /\  A  C_  X )  ->  (
( cls `  K
) `  ( F " A ) )  C_  ( F " ( ( cls `  J ) `
 A ) ) )
9 hmeocn 19446 . . 3  |-  ( F  e.  ( J Homeo K )  ->  F  e.  ( J  Cn  K
) )
102cnclsi 18989 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  A  C_  X )  -> 
( F " (
( cls `  J
) `  A )
)  C_  ( ( cls `  K ) `  ( F " A ) ) )
119, 10sylan 471 . 2  |-  ( ( F  e.  ( J
Homeo K )  /\  A  C_  X )  ->  ( F " ( ( cls `  J ) `  A
) )  C_  (
( cls `  K
) `  ( F " A ) ) )
128, 11eqssd 3468 1  |-  ( ( F  e.  ( J
Homeo K )  /\  A  C_  X )  ->  (
( cls `  K
) `  ( F " A ) )  =  ( F " (
( cls `  J
) `  A )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758    C_ wss 3423   U.cuni 4186   `'ccnv 4934   "cima 4938   ` cfv 5513  (class class class)co 6187   clsccl 18735    Cn ccn 18941   Homeochmeo 19439
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4498  ax-sep 4508  ax-nul 4516  ax-pow 4565  ax-pr 4626  ax-un 6469
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3067  df-sbc 3282  df-csb 3384  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-nul 3733  df-if 3887  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4187  df-int 4224  df-iun 4268  df-iin 4269  df-br 4388  df-opab 4446  df-mpt 4447  df-id 4731  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-rn 4946  df-res 4947  df-ima 4948  df-iota 5476  df-fun 5515  df-fn 5516  df-f 5517  df-f1 5518  df-fo 5519  df-f1o 5520  df-fv 5521  df-ov 6190  df-oprab 6191  df-mpt2 6192  df-map 7313  df-top 18616  df-topon 18619  df-cld 18736  df-cls 18738  df-cn 18944  df-hmeo 19441
This theorem is referenced by:  reghmph  19479  nrmhmph  19480  snclseqg  19799
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