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Theorem cnclsi 19958
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 19926 . . . . 5  |-  ( F  e.  ( J  Cn  K )  ->  J  e.  Top )
21adantr 463 . . . 4  |-  ( ( F  e.  ( J  Cn  K )  /\  S  C_  X )  ->  J  e.  Top )
3 cnvimass 5298 . . . . 5  |-  ( `' F " ( F
" S ) ) 
C_  dom  F
4 cnclsi.1 . . . . . . . 8  |-  X  = 
U. J
5 eqid 2402 . . . . . . . 8  |-  U. K  =  U. K
64, 5cnf 19932 . . . . . . 7  |-  ( F  e.  ( J  Cn  K )  ->  F : X --> U. K )
76adantr 463 . . . . . 6  |-  ( ( F  e.  ( J  Cn  K )  /\  S  C_  X )  ->  F : X --> U. K
)
8 fdm 5674 . . . . . 6  |-  ( F : X --> U. K  ->  dom  F  =  X )
97, 8syl 17 . . . . 5  |-  ( ( F  e.  ( J  Cn  K )  /\  S  C_  X )  ->  dom  F  =  X )
103, 9syl5sseq 3489 . . . 4  |-  ( ( F  e.  ( J  Cn  K )  /\  S  C_  X )  -> 
( `' F "
( F " S
) )  C_  X
)
11 simpr 459 . . . . . . 7  |-  ( ( F  e.  ( J  Cn  K )  /\  S  C_  X )  ->  S  C_  X )
1211, 9sseqtr4d 3478 . . . . . 6  |-  ( ( F  e.  ( J  Cn  K )  /\  S  C_  X )  ->  S  C_  dom  F )
13 dfss1 3643 . . . . . 6  |-  ( S 
C_  dom  F  <->  ( dom  F  i^i  S )  =  S )
1412, 13sylib 196 . . . . 5  |-  ( ( F  e.  ( J  Cn  K )  /\  S  C_  X )  -> 
( dom  F  i^i  S )  =  S )
15 dminss 5359 . . . . 5  |-  ( dom 
F  i^i  S )  C_  ( `' F "
( F " S
) )
1614, 15syl6eqssr 3492 . . . 4  |-  ( ( F  e.  ( J  Cn  K )  /\  S  C_  X )  ->  S  C_  ( `' F " ( F " S
) ) )
174clsss 19739 . . . 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 1230 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  S  C_  X )  -> 
( ( cls `  J
) `  S )  C_  ( ( cls `  J
) `  ( `' F " ( F " S ) ) ) )
19 imassrn 5289 . . . . 5  |-  ( F
" S )  C_  ran  F
20 frn 5676 . . . . . 6  |-  ( F : X --> U. K  ->  ran  F  C_  U. K
)
217, 20syl 17 . . . . 5  |-  ( ( F  e.  ( J  Cn  K )  /\  S  C_  X )  ->  ran  F  C_  U. K )
2219, 21syl5ss 3452 . . . 4  |-  ( ( F  e.  ( J  Cn  K )  /\  S  C_  X )  -> 
( F " S
)  C_  U. K )
235cncls2i 19956 . . . 4  |-  ( ( F  e.  ( J  Cn  K )  /\  ( F " S ) 
C_  U. K )  -> 
( ( cls `  J
) `  ( `' F " ( F " S ) ) ) 
C_  ( `' F " ( ( cls `  K
) `  ( F " S ) ) ) )
2422, 23syldan 468 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  S  C_  X )  -> 
( ( cls `  J
) `  ( `' F " ( F " S ) ) ) 
C_  ( `' F " ( ( cls `  K
) `  ( F " S ) ) ) )
2518, 24sstrd 3451 . 2  |-  ( ( F  e.  ( J  Cn  K )  /\  S  C_  X )  -> 
( ( cls `  J
) `  S )  C_  ( `' F "
( ( cls `  K
) `  ( F " S ) ) ) )
26 ffun 5672 . . . 4  |-  ( F : X --> U. K  ->  Fun  F )
277, 26syl 17 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  S  C_  X )  ->  Fun  F )
284clsss3 19744 . . . . 5  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( cls `  J
) `  S )  C_  X )
291, 28sylan 469 . . . 4  |-  ( ( F  e.  ( J  Cn  K )  /\  S  C_  X )  -> 
( ( cls `  J
) `  S )  C_  X )
3029, 9sseqtr4d 3478 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  S  C_  X )  -> 
( ( cls `  J
) `  S )  C_ 
dom  F )
31 funimass3 5937 . . 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 659 . 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 232 1  |-  ( ( F  e.  ( J  Cn  K )  /\  S  C_  X )  -> 
( F " (
( cls `  J
) `  S )
)  C_  ( ( cls `  K ) `  ( F " S ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1405    e. wcel 1842    i^i cin 3412    C_ wss 3413   U.cuni 4190   `'ccnv 4941   dom cdm 4942   ran crn 4943   "cima 4945   Fun wfun 5519   -->wf 5521   ` cfv 5525  (class class class)co 6234   Topctop 19578   clsccl 19703    Cn ccn 19910
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-iin 4273  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-ov 6237  df-oprab 6238  df-mpt2 6239  df-map 7379  df-top 19583  df-topon 19586  df-cld 19704  df-cls 19706  df-cn 19913
This theorem is referenced by:  cncls  19960  hmeocls  20453  clsnsg  20792
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