MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cnntri Structured version   Unicode version

Theorem cnntri 17340
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 17309 . . 3  |-  ( F  e.  ( J  Cn  K )  ->  J  e.  Top )
21adantr 453 . 2  |-  ( ( F  e.  ( J  Cn  K )  /\  S  C_  Y )  ->  J  e.  Top )
3 cnvimass 5227 . . 3  |-  ( `' F " S ) 
C_  dom  F
4 eqid 2438 . . . . . 6  |-  U. J  =  U. J
5 cncls2i.1 . . . . . 6  |-  Y  = 
U. K
64, 5cnf 17315 . . . . 5  |-  ( F  e.  ( J  Cn  K )  ->  F : U. J --> Y )
7 fdm 5598 . . . . 5  |-  ( F : U. J --> Y  ->  dom  F  =  U. J
)
86, 7syl 16 . . . 4  |-  ( F  e.  ( J  Cn  K )  ->  dom  F  =  U. J )
98adantr 453 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  S  C_  Y )  ->  dom  F  =  U. J
)
103, 9syl5sseq 3398 . 2  |-  ( ( F  e.  ( J  Cn  K )  /\  S  C_  Y )  -> 
( `' F " S )  C_  U. J
)
11 cntop2 17310 . . . 4  |-  ( F  e.  ( J  Cn  K )  ->  K  e.  Top )
125ntropn 17118 . . . 4  |-  ( ( K  e.  Top  /\  S  C_  Y )  -> 
( ( int `  K
) `  S )  e.  K )
1311, 12sylan 459 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  S  C_  Y )  -> 
( ( int `  K
) `  S )  e.  K )
14 cnima 17334 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  ( ( int `  K
) `  S )  e.  K )  ->  ( `' F " ( ( int `  K ) `
 S ) )  e.  J )
1513, 14syldan 458 . 2  |-  ( ( F  e.  ( J  Cn  K )  /\  S  C_  Y )  -> 
( `' F "
( ( int `  K
) `  S )
)  e.  J )
165ntrss2 17126 . . . 4  |-  ( ( K  e.  Top  /\  S  C_  Y )  -> 
( ( int `  K
) `  S )  C_  S )
1711, 16sylan 459 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  S  C_  Y )  -> 
( ( int `  K
) `  S )  C_  S )
18 imass2 5243 . . 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 17127 . 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 1186 1  |-  ( ( F  e.  ( J  Cn  K )  /\  S  C_  Y )  -> 
( `' F "
( ( int `  K
) `  S )
)  C_  ( ( int `  J ) `  ( `' F " S ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726    C_ wss 3322   U.cuni 4017   `'ccnv 4880   dom cdm 4881   "cima 4884   -->wf 5453   ` cfv 5457  (class class class)co 6084   Topctop 16963   intcnt 17086    Cn ccn 17293
This theorem is referenced by:  cnntr  17344  hmeontr  17806
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-map 7023  df-top 16968  df-topon 16971  df-ntr 17089  df-cn 17296
  Copyright terms: Public domain W3C validator