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Theorem cncls2i 19941
Description: Property of the preimage of a closure. (Contributed by Mario Carneiro, 25-Aug-2015.)
Hypothesis
Ref Expression
cncls2i.1  |-  Y  = 
U. K
Assertion
Ref Expression
cncls2i  |-  ( ( F  e.  ( J  Cn  K )  /\  S  C_  Y )  -> 
( ( cls `  J
) `  ( `' F " S ) ) 
C_  ( `' F " ( ( cls `  K
) `  S )
) )

Proof of Theorem cncls2i
StepHypRef Expression
1 cntop2 19912 . . . 4  |-  ( F  e.  ( J  Cn  K )  ->  K  e.  Top )
2 cncls2i.1 . . . . 5  |-  Y  = 
U. K
32clscld 19718 . . . 4  |-  ( ( K  e.  Top  /\  S  C_  Y )  -> 
( ( cls `  K
) `  S )  e.  ( Clsd `  K
) )
41, 3sylan 469 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  S  C_  Y )  -> 
( ( cls `  K
) `  S )  e.  ( Clsd `  K
) )
5 cnclima 19939 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  ( ( cls `  K
) `  S )  e.  ( Clsd `  K
) )  ->  ( `' F " ( ( cls `  K ) `
 S ) )  e.  ( Clsd `  J
) )
64, 5syldan 468 . 2  |-  ( ( F  e.  ( J  Cn  K )  /\  S  C_  Y )  -> 
( `' F "
( ( cls `  K
) `  S )
)  e.  ( Clsd `  J ) )
72sscls 19727 . . . 4  |-  ( ( K  e.  Top  /\  S  C_  Y )  ->  S  C_  ( ( cls `  K ) `  S
) )
81, 7sylan 469 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  S  C_  Y )  ->  S  C_  ( ( cls `  K ) `  S
) )
9 imass2 5360 . . 3  |-  ( S 
C_  ( ( cls `  K ) `  S
)  ->  ( `' F " S )  C_  ( `' F " ( ( cls `  K ) `
 S ) ) )
108, 9syl 16 . 2  |-  ( ( F  e.  ( J  Cn  K )  /\  S  C_  Y )  -> 
( `' F " S )  C_  ( `' F " ( ( cls `  K ) `
 S ) ) )
11 eqid 2454 . . 3  |-  U. J  =  U. J
1211clsss2 19743 . 2  |-  ( ( ( `' F "
( ( cls `  K
) `  S )
)  e.  ( Clsd `  J )  /\  ( `' F " S ) 
C_  ( `' F " ( ( cls `  K
) `  S )
) )  ->  (
( cls `  J
) `  ( `' F " S ) ) 
C_  ( `' F " ( ( cls `  K
) `  S )
) )
136, 10, 12syl2anc 659 1  |-  ( ( F  e.  ( J  Cn  K )  /\  S  C_  Y )  -> 
( ( cls `  J
) `  ( `' F " S ) ) 
C_  ( `' F " ( ( cls `  K
) `  S )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823    C_ wss 3461   U.cuni 4235   `'ccnv 4987   "cima 4991   ` cfv 5570  (class class class)co 6270   Topctop 19564   Clsdccld 19687   clsccl 19689    Cn ccn 19895
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-iin 4318  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-map 7414  df-top 19569  df-topon 19572  df-cld 19690  df-cls 19692  df-cn 19898
This theorem is referenced by:  cnclsi  19943  cncls2  19944  imasncls  20362  hmeocls  20438  clssubg  20776
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