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Theorem cnclsi 17001
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 16970 . . . . 5  |-  ( F  e.  ( J  Cn  K )  ->  J  e.  Top )
21adantr 451 . . . 4  |-  ( ( F  e.  ( J  Cn  K )  /\  S  C_  X )  ->  J  e.  Top )
3 cnvimass 5033 . . . . 5  |-  ( `' F " ( F
" S ) ) 
C_  dom  F
4 cnclsi.1 . . . . . . . 8  |-  X  = 
U. J
5 eqid 2283 . . . . . . . 8  |-  U. K  =  U. K
64, 5cnf 16976 . . . . . . 7  |-  ( F  e.  ( J  Cn  K )  ->  F : X --> U. K )
76adantr 451 . . . . . 6  |-  ( ( F  e.  ( J  Cn  K )  /\  S  C_  X )  ->  F : X --> U. K
)
8 fdm 5393 . . . . . 6  |-  ( F : X --> U. K  ->  dom  F  =  X )
97, 8syl 15 . . . . 5  |-  ( ( F  e.  ( J  Cn  K )  /\  S  C_  X )  ->  dom  F  =  X )
103, 9syl5sseq 3226 . . . 4  |-  ( ( F  e.  ( J  Cn  K )  /\  S  C_  X )  -> 
( `' F "
( F " S
) )  C_  X
)
11 dminss 5095 . . . . 5  |-  ( dom 
F  i^i  S )  C_  ( `' F "
( F " S
) )
12 simpr 447 . . . . . . . 8  |-  ( ( F  e.  ( J  Cn  K )  /\  S  C_  X )  ->  S  C_  X )
1312, 9sseqtr4d 3215 . . . . . . 7  |-  ( ( F  e.  ( J  Cn  K )  /\  S  C_  X )  ->  S  C_  dom  F )
14 dfss1 3373 . . . . . . 7  |-  ( S 
C_  dom  F  <->  ( dom  F  i^i  S )  =  S )
1513, 14sylib 188 . . . . . 6  |-  ( ( F  e.  ( J  Cn  K )  /\  S  C_  X )  -> 
( dom  F  i^i  S )  =  S )
1615sseq1d 3205 . . . . 5  |-  ( ( F  e.  ( J  Cn  K )  /\  S  C_  X )  -> 
( ( dom  F  i^i  S )  C_  ( `' F " ( F
" S ) )  <-> 
S  C_  ( `' F " ( F " S ) ) ) )
1711, 16mpbii 202 . . . 4  |-  ( ( F  e.  ( J  Cn  K )  /\  S  C_  X )  ->  S  C_  ( `' F " ( F " S
) ) )
184clsss 16791 . . . 4  |-  ( ( J  e.  Top  /\  ( `' F " ( F
" S ) ) 
C_  X  /\  S  C_  ( `' F "
( F " S
) ) )  -> 
( ( cls `  J
) `  S )  C_  ( ( cls `  J
) `  ( `' F " ( F " S ) ) ) )
192, 10, 17, 18syl3anc 1182 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  S  C_  X )  -> 
( ( cls `  J
) `  S )  C_  ( ( cls `  J
) `  ( `' F " ( F " S ) ) ) )
20 imassrn 5025 . . . . 5  |-  ( F
" S )  C_  ran  F
21 frn 5395 . . . . . 6  |-  ( F : X --> U. K  ->  ran  F  C_  U. K
)
227, 21syl 15 . . . . 5  |-  ( ( F  e.  ( J  Cn  K )  /\  S  C_  X )  ->  ran  F  C_  U. K )
2320, 22syl5ss 3190 . . . 4  |-  ( ( F  e.  ( J  Cn  K )  /\  S  C_  X )  -> 
( F " S
)  C_  U. K )
245cncls2i 16999 . . . 4  |-  ( ( F  e.  ( J  Cn  K )  /\  ( F " S ) 
C_  U. K )  -> 
( ( cls `  J
) `  ( `' F " ( F " S ) ) ) 
C_  ( `' F " ( ( cls `  K
) `  ( F " S ) ) ) )
2523, 24syldan 456 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  S  C_  X )  -> 
( ( cls `  J
) `  ( `' F " ( F " S ) ) ) 
C_  ( `' F " ( ( cls `  K
) `  ( F " S ) ) ) )
2619, 25sstrd 3189 . 2  |-  ( ( F  e.  ( J  Cn  K )  /\  S  C_  X )  -> 
( ( cls `  J
) `  S )  C_  ( `' F "
( ( cls `  K
) `  ( F " S ) ) ) )
27 ffun 5391 . . . 4  |-  ( F : X --> U. K  ->  Fun  F )
287, 27syl 15 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  S  C_  X )  ->  Fun  F )
294clsss3 16796 . . . . 5  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( cls `  J
) `  S )  C_  X )
301, 29sylan 457 . . . 4  |-  ( ( F  e.  ( J  Cn  K )  /\  S  C_  X )  -> 
( ( cls `  J
) `  S )  C_  X )
3130, 9sseqtr4d 3215 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  S  C_  X )  -> 
( ( cls `  J
) `  S )  C_ 
dom  F )
32 funimass3 5641 . . 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 ) ) ) ) )
3328, 31, 32syl2anc 642 . 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 ) ) ) ) )
3426, 33mpbird 223 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 176    /\ wa 358    = wceq 1623    e. wcel 1684    i^i cin 3151    C_ wss 3152   U.cuni 3827   `'ccnv 4688   dom cdm 4689   ran crn 4690   "cima 4692   Fun wfun 5249   -->wf 5251   ` cfv 5255  (class class class)co 5858   Topctop 16631   clsccl 16755    Cn ccn 16954
This theorem is referenced by:  cncls  17003  hmeocls  17459  clsnsg  17792
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-map 6774  df-top 16636  df-topon 16639  df-cld 16756  df-cls 16758  df-cn 16957
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