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Theorem cnntri 17293
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 17262 . . 3  |-  ( F  e.  ( J  Cn  K )  ->  J  e.  Top )
21adantr 452 . 2  |-  ( ( F  e.  ( J  Cn  K )  /\  S  C_  Y )  ->  J  e.  Top )
3 cnvimass 5187 . . 3  |-  ( `' F " S ) 
C_  dom  F
4 eqid 2408 . . . . . 6  |-  U. J  =  U. J
5 cncls2i.1 . . . . . 6  |-  Y  = 
U. K
64, 5cnf 17268 . . . . 5  |-  ( F  e.  ( J  Cn  K )  ->  F : U. J --> Y )
7 fdm 5558 . . . . 5  |-  ( F : U. J --> Y  ->  dom  F  =  U. J
)
86, 7syl 16 . . . 4  |-  ( F  e.  ( J  Cn  K )  ->  dom  F  =  U. J )
98adantr 452 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  S  C_  Y )  ->  dom  F  =  U. J
)
103, 9syl5sseq 3360 . 2  |-  ( ( F  e.  ( J  Cn  K )  /\  S  C_  Y )  -> 
( `' F " S )  C_  U. J
)
11 cntop2 17263 . . . 4  |-  ( F  e.  ( J  Cn  K )  ->  K  e.  Top )
125ntropn 17072 . . . 4  |-  ( ( K  e.  Top  /\  S  C_  Y )  -> 
( ( int `  K
) `  S )  e.  K )
1311, 12sylan 458 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  S  C_  Y )  -> 
( ( int `  K
) `  S )  e.  K )
14 cnima 17287 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  ( ( int `  K
) `  S )  e.  K )  ->  ( `' F " ( ( int `  K ) `
 S ) )  e.  J )
1513, 14syldan 457 . 2  |-  ( ( F  e.  ( J  Cn  K )  /\  S  C_  Y )  -> 
( `' F "
( ( int `  K
) `  S )
)  e.  J )
165ntrss2 17080 . . . 4  |-  ( ( K  e.  Top  /\  S  C_  Y )  -> 
( ( int `  K
) `  S )  C_  S )
1711, 16sylan 458 . . 3  |-  ( ( F  e.  ( J  Cn  K )  /\  S  C_  Y )  -> 
( ( int `  K
) `  S )  C_  S )
18 imass2 5203 . . 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 17081 . 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 1185 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 359    = wceq 1649    e. wcel 1721    C_ wss 3284   U.cuni 3979   `'ccnv 4840   dom cdm 4841   "cima 4844   -->wf 5413   ` cfv 5417  (class class class)co 6044   Topctop 16917   intcnt 17040    Cn ccn 17246
This theorem is referenced by:  cnntr  17297  hmeontr  17758
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-reu 2677  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-map 6983  df-top 16922  df-topon 16925  df-ntr 17043  df-cn 17249
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