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Theorem cnclsi 19018
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 18986 . . . . 5  |-  ( F  e.  ( J  Cn  K )  ->  J  e.  Top )
21adantr 465 . . . 4  |-  ( ( F  e.  ( J  Cn  K )  /\  S  C_  X )  ->  J  e.  Top )
3 cnvimass 5300 . . . . 5  |-  ( `' F " ( F
" S ) ) 
C_  dom  F
4 cnclsi.1 . . . . . . . 8  |-  X  = 
U. J
5 eqid 2454 . . . . . . . 8  |-  U. K  =  U. K
64, 5cnf 18992 . . . . . . 7  |-  ( F  e.  ( J  Cn  K )  ->  F : X --> U. K )
76adantr 465 . . . . . 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 16 . . . . 5  |-  ( ( F  e.  ( J  Cn  K )  /\  S  C_  X )  ->  dom  F  =  X )
103, 9syl5sseq 3515 . . . 4  |-  ( ( F  e.  ( J  Cn  K )  /\  S  C_  X )  -> 
( `' F "
( F " S
) )  C_  X
)
11 simpr 461 . . . . . . 7  |-  ( ( F  e.  ( J  Cn  K )  /\  S  C_  X )  ->  S  C_  X )
1211, 9sseqtr4d 3504 . . . . . 6  |-  ( ( F  e.  ( J  Cn  K )  /\  S  C_  X )  ->  S  C_  dom  F )
13 dfss1 3666 . . . . . 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 5362 . . . . 5  |-  ( dom 
F  i^i  S )  C_  ( `' F "
( F " S
) )
1614, 15syl6eqssr 3518 . . . 4  |-  ( ( F  e.  ( J  Cn  K )  /\  S  C_  X )  ->  S  C_  ( `' F " ( F " S
) ) )
174clsss 18800 . . . 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 1219 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  S  C_  X )  -> 
( ( cls `  J
) `  S )  C_  ( ( cls `  J
) `  ( `' F " ( F " S ) ) ) )
19 imassrn 5291 . . . . 5  |-  ( F
" S )  C_  ran  F
20 frn 5676 . . . . . 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 3478 . . . 4  |-  ( ( F  e.  ( J  Cn  K )  /\  S  C_  X )  -> 
( F " S
)  C_  U. K )
235cncls2i 19016 . . . 4  |-  ( ( F  e.  ( J  Cn  K )  /\  ( F " S ) 
C_  U. K )  -> 
( ( cls `  J
) `  ( `' F " ( F " S ) ) ) 
C_  ( `' F " ( ( cls `  K
) `  ( F " S ) ) ) )
2422, 23syldan 470 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  S  C_  X )  -> 
( ( cls `  J
) `  ( `' F " ( F " S ) ) ) 
C_  ( `' F " ( ( cls `  K
) `  ( F " S ) ) ) )
2518, 24sstrd 3477 . 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 16 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  S  C_  X )  ->  Fun  F )
284clsss3 18805 . . . . 5  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( cls `  J
) `  S )  C_  X )
291, 28sylan 471 . . . 4  |-  ( ( F  e.  ( J  Cn  K )  /\  S  C_  X )  -> 
( ( cls `  J
) `  S )  C_  X )
3029, 9sseqtr4d 3504 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  S  C_  X )  -> 
( ( cls `  J
) `  S )  C_ 
dom  F )
31 funimass3 5931 . . 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 661 . 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 369    = wceq 1370    e. wcel 1758    i^i cin 3438    C_ wss 3439   U.cuni 4202   `'ccnv 4950   dom cdm 4951   ran crn 4952   "cima 4954   Fun wfun 5523   -->wf 5525   ` cfv 5529  (class class class)co 6203   Topctop 18640   clsccl 18764    Cn ccn 18970
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-iin 4285  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-map 7329  df-top 18645  df-topon 18648  df-cld 18765  df-cls 18767  df-cn 18973
This theorem is referenced by:  cncls  19020  hmeocls  19483  clsnsg  19822
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