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Theorem cnntri 18880
Description: Property of the preimage of an interior. (Contributed by Mario Carneiro, 25-Aug-2015.)
Hypothesis
Ref Expression
cncls2i.1  |-  Y  = 
U. K
Assertion
Ref Expression
cnntri  |-  ( ( F  e.  ( J  Cn  K )  /\  S  C_  Y )  -> 
( `' F "
( ( int `  K
) `  S )
)  C_  ( ( int `  J ) `  ( `' F " S ) ) )

Proof of Theorem cnntri
StepHypRef Expression
1 cntop1 18849 . . 3  |-  ( F  e.  ( J  Cn  K )  ->  J  e.  Top )
21adantr 465 . 2  |-  ( ( F  e.  ( J  Cn  K )  /\  S  C_  Y )  ->  J  e.  Top )
3 cnvimass 5194 . . 3  |-  ( `' F " S ) 
C_  dom  F
4 eqid 2443 . . . . . 6  |-  U. J  =  U. J
5 cncls2i.1 . . . . . 6  |-  Y  = 
U. K
64, 5cnf 18855 . . . . 5  |-  ( F  e.  ( J  Cn  K )  ->  F : U. J --> Y )
7 fdm 5568 . . . . 5  |-  ( F : U. J --> Y  ->  dom  F  =  U. J
)
86, 7syl 16 . . . 4  |-  ( F  e.  ( J  Cn  K )  ->  dom  F  =  U. J )
98adantr 465 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  S  C_  Y )  ->  dom  F  =  U. J
)
103, 9syl5sseq 3409 . 2  |-  ( ( F  e.  ( J  Cn  K )  /\  S  C_  Y )  -> 
( `' F " S )  C_  U. J
)
11 cntop2 18850 . . . 4  |-  ( F  e.  ( J  Cn  K )  ->  K  e.  Top )
125ntropn 18658 . . . 4  |-  ( ( K  e.  Top  /\  S  C_  Y )  -> 
( ( int `  K
) `  S )  e.  K )
1311, 12sylan 471 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  S  C_  Y )  -> 
( ( int `  K
) `  S )  e.  K )
14 cnima 18874 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  ( ( int `  K
) `  S )  e.  K )  ->  ( `' F " ( ( int `  K ) `
 S ) )  e.  J )
1513, 14syldan 470 . 2  |-  ( ( F  e.  ( J  Cn  K )  /\  S  C_  Y )  -> 
( `' F "
( ( int `  K
) `  S )
)  e.  J )
165ntrss2 18666 . . . 4  |-  ( ( K  e.  Top  /\  S  C_  Y )  -> 
( ( int `  K
) `  S )  C_  S )
1711, 16sylan 471 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  S  C_  Y )  -> 
( ( int `  K
) `  S )  C_  S )
18 imass2 5209 . . 3  |-  ( ( ( int `  K
) `  S )  C_  S  ->  ( `' F " ( ( int `  K ) `  S
) )  C_  ( `' F " S ) )
1917, 18syl 16 . 2  |-  ( ( F  e.  ( J  Cn  K )  /\  S  C_  Y )  -> 
( `' F "
( ( int `  K
) `  S )
)  C_  ( `' F " S ) )
204ssntr 18667 . 2  |-  ( ( ( J  e.  Top  /\  ( `' F " S )  C_  U. J
)  /\  ( ( `' F " ( ( int `  K ) `
 S ) )  e.  J  /\  ( `' F " ( ( int `  K ) `
 S ) ) 
C_  ( `' F " S ) ) )  ->  ( `' F " ( ( int `  K
) `  S )
)  C_  ( ( int `  J ) `  ( `' F " S ) ) )
212, 10, 15, 19, 20syl22anc 1219 1  |-  ( ( F  e.  ( J  Cn  K )  /\  S  C_  Y )  -> 
( `' F "
( ( int `  K
) `  S )
)  C_  ( ( int `  J ) `  ( `' F " S ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756    C_ wss 3333   U.cuni 4096   `'ccnv 4844   dom cdm 4845   "cima 4848   -->wf 5419   ` cfv 5423  (class class class)co 6096   Topctop 18503   intcnt 18626    Cn ccn 18833
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-map 7221  df-top 18508  df-topon 18511  df-ntr 18629  df-cn 18836
This theorem is referenced by:  cnntr  18884  hmeontr  19347
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