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Theorem cnclsi 17336
Description: Property of the image of a closure. (Contributed by Mario Carneiro, 25-Aug-2015.)
Hypothesis
Ref Expression
cnclsi.1  |-  X  = 
U. J
Assertion
Ref Expression
cnclsi  |-  ( ( F  e.  ( J  Cn  K )  /\  S  C_  X )  -> 
( F " (
( cls `  J
) `  S )
)  C_  ( ( cls `  K ) `  ( F " S ) ) )

Proof of Theorem cnclsi
StepHypRef Expression
1 cntop1 17304 . . . . 5  |-  ( F  e.  ( J  Cn  K )  ->  J  e.  Top )
21adantr 452 . . . 4  |-  ( ( F  e.  ( J  Cn  K )  /\  S  C_  X )  ->  J  e.  Top )
3 cnvimass 5224 . . . . 5  |-  ( `' F " ( F
" S ) ) 
C_  dom  F
4 cnclsi.1 . . . . . . . 8  |-  X  = 
U. J
5 eqid 2436 . . . . . . . 8  |-  U. K  =  U. K
64, 5cnf 17310 . . . . . . 7  |-  ( F  e.  ( J  Cn  K )  ->  F : X --> U. K )
76adantr 452 . . . . . 6  |-  ( ( F  e.  ( J  Cn  K )  /\  S  C_  X )  ->  F : X --> U. K
)
8 fdm 5595 . . . . . 6  |-  ( F : X --> U. K  ->  dom  F  =  X )
97, 8syl 16 . . . . 5  |-  ( ( F  e.  ( J  Cn  K )  /\  S  C_  X )  ->  dom  F  =  X )
103, 9syl5sseq 3396 . . . 4  |-  ( ( F  e.  ( J  Cn  K )  /\  S  C_  X )  -> 
( `' F "
( F " S
) )  C_  X
)
11 simpr 448 . . . . . . 7  |-  ( ( F  e.  ( J  Cn  K )  /\  S  C_  X )  ->  S  C_  X )
1211, 9sseqtr4d 3385 . . . . . 6  |-  ( ( F  e.  ( J  Cn  K )  /\  S  C_  X )  ->  S  C_  dom  F )
13 dfss1 3545 . . . . . 6  |-  ( S 
C_  dom  F  <->  ( dom  F  i^i  S )  =  S )
1412, 13sylib 189 . . . . 5  |-  ( ( F  e.  ( J  Cn  K )  /\  S  C_  X )  -> 
( dom  F  i^i  S )  =  S )
15 dminss 5286 . . . . 5  |-  ( dom 
F  i^i  S )  C_  ( `' F "
( F " S
) )
1614, 15syl6eqssr 3399 . . . 4  |-  ( ( F  e.  ( J  Cn  K )  /\  S  C_  X )  ->  S  C_  ( `' F " ( F " S
) ) )
174clsss 17118 . . . 4  |-  ( ( J  e.  Top  /\  ( `' F " ( F
" S ) ) 
C_  X  /\  S  C_  ( `' F "
( F " S
) ) )  -> 
( ( cls `  J
) `  S )  C_  ( ( cls `  J
) `  ( `' F " ( F " S ) ) ) )
182, 10, 16, 17syl3anc 1184 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  S  C_  X )  -> 
( ( cls `  J
) `  S )  C_  ( ( cls `  J
) `  ( `' F " ( F " S ) ) ) )
19 imassrn 5216 . . . . 5  |-  ( F
" S )  C_  ran  F
20 frn 5597 . . . . . 6  |-  ( F : X --> U. K  ->  ran  F  C_  U. K
)
217, 20syl 16 . . . . 5  |-  ( ( F  e.  ( J  Cn  K )  /\  S  C_  X )  ->  ran  F  C_  U. K )
2219, 21syl5ss 3359 . . . 4  |-  ( ( F  e.  ( J  Cn  K )  /\  S  C_  X )  -> 
( F " S
)  C_  U. K )
235cncls2i 17334 . . . 4  |-  ( ( F  e.  ( J  Cn  K )  /\  ( F " S ) 
C_  U. K )  -> 
( ( cls `  J
) `  ( `' F " ( F " S ) ) ) 
C_  ( `' F " ( ( cls `  K
) `  ( F " S ) ) ) )
2422, 23syldan 457 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  S  C_  X )  -> 
( ( cls `  J
) `  ( `' F " ( F " S ) ) ) 
C_  ( `' F " ( ( cls `  K
) `  ( F " S ) ) ) )
2518, 24sstrd 3358 . 2  |-  ( ( F  e.  ( J  Cn  K )  /\  S  C_  X )  -> 
( ( cls `  J
) `  S )  C_  ( `' F "
( ( cls `  K
) `  ( F " S ) ) ) )
26 ffun 5593 . . . 4  |-  ( F : X --> U. K  ->  Fun  F )
277, 26syl 16 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  S  C_  X )  ->  Fun  F )
284clsss3 17123 . . . . 5  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( cls `  J
) `  S )  C_  X )
291, 28sylan 458 . . . 4  |-  ( ( F  e.  ( J  Cn  K )  /\  S  C_  X )  -> 
( ( cls `  J
) `  S )  C_  X )
3029, 9sseqtr4d 3385 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  S  C_  X )  -> 
( ( cls `  J
) `  S )  C_ 
dom  F )
31 funimass3 5846 . . 3  |-  ( ( Fun  F  /\  (
( cls `  J
) `  S )  C_ 
dom  F )  -> 
( ( F "
( ( cls `  J
) `  S )
)  C_  ( ( cls `  K ) `  ( F " S ) )  <->  ( ( cls `  J ) `  S
)  C_  ( `' F " ( ( cls `  K ) `  ( F " S ) ) ) ) )
3227, 30, 31syl2anc 643 . 2  |-  ( ( F  e.  ( J  Cn  K )  /\  S  C_  X )  -> 
( ( F "
( ( cls `  J
) `  S )
)  C_  ( ( cls `  K ) `  ( F " S ) )  <->  ( ( cls `  J ) `  S
)  C_  ( `' F " ( ( cls `  K ) `  ( F " S ) ) ) ) )
3325, 32mpbird 224 1  |-  ( ( F  e.  ( J  Cn  K )  /\  S  C_  X )  -> 
( F " (
( cls `  J
) `  S )
)  C_  ( ( cls `  K ) `  ( F " S ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725    i^i cin 3319    C_ wss 3320   U.cuni 4015   `'ccnv 4877   dom cdm 4878   ran crn 4879   "cima 4881   Fun wfun 5448   -->wf 5450   ` cfv 5454  (class class class)co 6081   Topctop 16958   clsccl 17082    Cn ccn 17288
This theorem is referenced by:  cncls  17338  hmeocls  17800  clsnsg  18139
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
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