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Theorem cnntri 19899
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 19868 . . 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 5367 . . 3  |-  ( `' F " S ) 
C_  dom  F
4 eqid 2457 . . . . . 6  |-  U. J  =  U. J
5 cncls2i.1 . . . . . 6  |-  Y  = 
U. K
64, 5cnf 19874 . . . . 5  |-  ( F  e.  ( J  Cn  K )  ->  F : U. J --> Y )
7 fdm 5741 . . . . 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 3547 . 2  |-  ( ( F  e.  ( J  Cn  K )  /\  S  C_  Y )  -> 
( `' F " S )  C_  U. J
)
11 cntop2 19869 . . . 4  |-  ( F  e.  ( J  Cn  K )  ->  K  e.  Top )
125ntropn 19677 . . . 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 19893 . . 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 19685 . . . 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 5382 . . 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 19686 . 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 1229 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 1395    e. wcel 1819    C_ wss 3471   U.cuni 4251   `'ccnv 5007   dom cdm 5008   "cima 5011   -->wf 5590   ` cfv 5594  (class class class)co 6296   Topctop 19521   intcnt 19645    Cn ccn 19852
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-map 7440  df-top 19526  df-topon 19529  df-ntr 19648  df-cn 19855
This theorem is referenced by:  cnntr  19903  hmeontr  20396
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