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Theorem cnclsi 17017
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 16986 . . . . 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 5049 . . . . 5  |-  ( `' F " ( F
" S ) ) 
C_  dom  F
4 cnclsi.1 . . . . . . . 8  |-  X  = 
U. J
5 eqid 2296 . . . . . . . 8  |-  U. K  =  U. K
64, 5cnf 16992 . . . . . . 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 5409 . . . . . 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 3239 . . . 4  |-  ( ( F  e.  ( J  Cn  K )  /\  S  C_  X )  -> 
( `' F "
( F " S
) )  C_  X
)
11 dminss 5111 . . . . 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 3228 . . . . . . 7  |-  ( ( F  e.  ( J  Cn  K )  /\  S  C_  X )  ->  S  C_  dom  F )
14 dfss1 3386 . . . . . . 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 3218 . . . . 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 16807 . . . 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 5041 . . . . 5  |-  ( F
" S )  C_  ran  F
21 frn 5411 . . . . . 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 3203 . . . 4  |-  ( ( F  e.  ( J  Cn  K )  /\  S  C_  X )  -> 
( F " S
)  C_  U. K )
245cncls2i 17015 . . . 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 3202 . 2  |-  ( ( F  e.  ( J  Cn  K )  /\  S  C_  X )  -> 
( ( cls `  J
) `  S )  C_  ( `' F "
( ( cls `  K
) `  ( F " S ) ) ) )
27 ffun 5407 . . . 4  |-  ( F : X --> U. K  ->  Fun  F )
287, 27syl 15 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  S  C_  X )  ->  Fun  F )
294clsss3 16812 . . . . 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 3228 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  S  C_  X )  -> 
( ( cls `  J
) `  S )  C_ 
dom  F )
32 funimass3 5657 . . 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 1632    e. wcel 1696    i^i cin 3164    C_ wss 3165   U.cuni 3843   `'ccnv 4704   dom cdm 4705   ran crn 4706   "cima 4708   Fun wfun 5265   -->wf 5267   ` cfv 5271  (class class class)co 5874   Topctop 16647   clsccl 16771    Cn ccn 16970
This theorem is referenced by:  cncls  17019  hmeocls  17475  clsnsg  17808
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-map 6790  df-top 16652  df-topon 16655  df-cld 16772  df-cls 16774  df-cn 16973
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