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Theorem hmeocls 20354
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 20347 . . . 4  |-  ( F  e.  ( J Homeo K )  ->  `' F  e.  ( K  Cn  J
) )
2 hmeoopn.1 . . . . 5  |-  X  = 
U. J
32cncls2i 19857 . . . 4  |-  ( ( `' F  e.  ( K  Cn  J )  /\  A  C_  X )  -> 
( ( cls `  K
) `  ( `' `' F " A ) )  C_  ( `' `' F " ( ( cls `  J ) `
 A ) ) )
41, 3sylan 469 . . 3  |-  ( ( F  e.  ( J
Homeo K )  /\  A  C_  X )  ->  (
( cls `  K
) `  ( `' `' F " A ) )  C_  ( `' `' F " ( ( cls `  J ) `
 A ) ) )
5 imacnvcnv 5380 . . . 4  |-  ( `' `' F " A )  =  ( F " A )
65fveq2i 5777 . . 3  |-  ( ( cls `  K ) `
 ( `' `' F " A ) )  =  ( ( cls `  K ) `  ( F " A ) )
7 imacnvcnv 5380 . . 3  |-  ( `' `' F " ( ( cls `  J ) `
 A ) )  =  ( F "
( ( cls `  J
) `  A )
)
84, 6, 73sstr3g 3457 . 2  |-  ( ( F  e.  ( J
Homeo K )  /\  A  C_  X )  ->  (
( cls `  K
) `  ( F " A ) )  C_  ( F " ( ( cls `  J ) `
 A ) ) )
9 hmeocn 20346 . . 3  |-  ( F  e.  ( J Homeo K )  ->  F  e.  ( J  Cn  K
) )
102cnclsi 19859 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  A  C_  X )  -> 
( F " (
( cls `  J
) `  A )
)  C_  ( ( cls `  K ) `  ( F " A ) ) )
119, 10sylan 469 . 2  |-  ( ( F  e.  ( J
Homeo K )  /\  A  C_  X )  ->  ( F " ( ( cls `  J ) `  A
) )  C_  (
( cls `  K
) `  ( F " A ) ) )
128, 11eqssd 3434 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 367    = wceq 1399    e. wcel 1826    C_ wss 3389   U.cuni 4163   `'ccnv 4912   "cima 4916   ` cfv 5496  (class class class)co 6196   clsccl 19604    Cn ccn 19811   Homeochmeo 20339
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-rep 4478  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-reu 2739  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-int 4200  df-iun 4245  df-iin 4246  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-map 7340  df-top 19484  df-topon 19487  df-cld 19605  df-cls 19607  df-cn 19814  df-hmeo 20341
This theorem is referenced by:  reghmph  20379  nrmhmph  20380  snclseqg  20699
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