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Theorem hmeoclda 28699
Description: Homeomorphisms preserve closedness. (Contributed by Jeff Hankins, 3-Jul-2009.) (Revised by Mario Carneiro, 3-Jun-2014.)
Assertion
Ref Expression
hmeoclda  |-  ( ( ( J  e.  Top  /\  K  e.  Top  /\  F  e.  ( J Homeo K ) )  /\  S  e.  ( Clsd `  J ) )  -> 
( F " S
)  e.  ( Clsd `  K ) )

Proof of Theorem hmeoclda
StepHypRef Expression
1 hmeocnvcn 19476 . . 3  |-  ( F  e.  ( J Homeo K )  ->  `' F  e.  ( K  Cn  J
) )
213ad2ant3 1011 . 2  |-  ( ( J  e.  Top  /\  K  e.  Top  /\  F  e.  ( J Homeo K ) )  ->  `' F  e.  ( K  Cn  J
) )
3 imacnvcnv 5414 . . 3  |-  ( `' `' F " S )  =  ( F " S )
4 cnclima 19014 . . 3  |-  ( ( `' F  e.  ( K  Cn  J )  /\  S  e.  ( Clsd `  J ) )  -> 
( `' `' F " S )  e.  (
Clsd `  K )
)
53, 4syl5eqelr 2547 . 2  |-  ( ( `' F  e.  ( K  Cn  J )  /\  S  e.  ( Clsd `  J ) )  -> 
( F " S
)  e.  ( Clsd `  K ) )
62, 5sylan 471 1  |-  ( ( ( J  e.  Top  /\  K  e.  Top  /\  F  e.  ( J Homeo K ) )  /\  S  e.  ( Clsd `  J ) )  -> 
( F " S
)  e.  ( Clsd `  K ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    e. wcel 1758   `'ccnv 4950   "cima 4954   ` cfv 5529  (class class class)co 6203   Topctop 18640   Clsdccld 18762    Cn ccn 18970   Homeochmeo 19468
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 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
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 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-fv 5537  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-map 7329  df-top 18645  df-topon 18648  df-cld 18765  df-cn 18973  df-hmeo 19470
This theorem is referenced by: (None)
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