ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  hmeontr Unicode version

Theorem hmeontr 12755
Description: Homeomorphisms preserve interiors. (Contributed by Mario Carneiro, 25-Aug-2015.)
Hypothesis
Ref Expression
hmeoopn.1  |-  X  = 
U. J
Assertion
Ref Expression
hmeontr  |-  ( ( F  e.  ( J
Homeo K )  /\  A  C_  X )  ->  (
( int `  K
) `  ( F " A ) )  =  ( F " (
( int `  J
) `  A )
) )

Proof of Theorem hmeontr
StepHypRef Expression
1 hmeocn 12747 . . . . . 6  |-  ( F  e.  ( J Homeo K )  ->  F  e.  ( J  Cn  K
) )
21adantr 274 . . . . 5  |-  ( ( F  e.  ( J
Homeo K )  /\  A  C_  X )  ->  F  e.  ( J  Cn  K
) )
3 imassrn 4940 . . . . . 6  |-  ( F
" A )  C_  ran  F
4 hmeoopn.1 . . . . . . . . 9  |-  X  = 
U. J
5 eqid 2157 . . . . . . . . 9  |-  U. K  =  U. K
64, 5hmeof1o 12751 . . . . . . . 8  |-  ( F  e.  ( J Homeo K )  ->  F : X
-1-1-onto-> U. K )
76adantr 274 . . . . . . 7  |-  ( ( F  e.  ( J
Homeo K )  /\  A  C_  X )  ->  F : X -1-1-onto-> U. K )
8 f1ofo 5422 . . . . . . 7  |-  ( F : X -1-1-onto-> U. K  ->  F : X -onto-> U. K )
9 forn 5396 . . . . . . 7  |-  ( F : X -onto-> U. K  ->  ran  F  =  U. K )
107, 8, 93syl 17 . . . . . 6  |-  ( ( F  e.  ( J
Homeo K )  /\  A  C_  X )  ->  ran  F  =  U. K )
113, 10sseqtrid 3178 . . . . 5  |-  ( ( F  e.  ( J
Homeo K )  /\  A  C_  X )  ->  ( F " A )  C_  U. K )
125cnntri 12666 . . . . 5  |-  ( ( F  e.  ( J  Cn  K )  /\  ( F " A ) 
C_  U. K )  -> 
( `' F "
( ( int `  K
) `  ( F " A ) ) ) 
C_  ( ( int `  J ) `  ( `' F " ( F
" A ) ) ) )
132, 11, 12syl2anc 409 . . . 4  |-  ( ( F  e.  ( J
Homeo K )  /\  A  C_  X )  ->  ( `' F " ( ( int `  K ) `
 ( F " A ) ) ) 
C_  ( ( int `  J ) `  ( `' F " ( F
" A ) ) ) )
14 f1of1 5414 . . . . . . 7  |-  ( F : X -1-1-onto-> U. K  ->  F : X -1-1-> U. K )
157, 14syl 14 . . . . . 6  |-  ( ( F  e.  ( J
Homeo K )  /\  A  C_  X )  ->  F : X -1-1-> U. K )
16 f1imacnv 5432 . . . . . 6  |-  ( ( F : X -1-1-> U. K  /\  A  C_  X
)  ->  ( `' F " ( F " A ) )  =  A )
1715, 16sylancom 417 . . . . 5  |-  ( ( F  e.  ( J
Homeo K )  /\  A  C_  X )  ->  ( `' F " ( F
" A ) )  =  A )
1817fveq2d 5473 . . . 4  |-  ( ( F  e.  ( J
Homeo K )  /\  A  C_  X )  ->  (
( int `  J
) `  ( `' F " ( F " A ) ) )  =  ( ( int `  J ) `  A
) )
1913, 18sseqtrd 3166 . . 3  |-  ( ( F  e.  ( J
Homeo K )  /\  A  C_  X )  ->  ( `' F " ( ( int `  K ) `
 ( F " A ) ) ) 
C_  ( ( int `  J ) `  A
) )
20 f1ofun 5417 . . . . 5  |-  ( F : X -1-1-onto-> U. K  ->  Fun  F )
217, 20syl 14 . . . 4  |-  ( ( F  e.  ( J
Homeo K )  /\  A  C_  X )  ->  Fun  F )
22 cntop2 12644 . . . . . . 7  |-  ( F  e.  ( J  Cn  K )  ->  K  e.  Top )
232, 22syl 14 . . . . . 6  |-  ( ( F  e.  ( J
Homeo K )  /\  A  C_  X )  ->  K  e.  Top )
245ntrss3 12565 . . . . . 6  |-  ( ( K  e.  Top  /\  ( F " A ) 
C_  U. K )  -> 
( ( int `  K
) `  ( F " A ) )  C_  U. K )
2523, 11, 24syl2anc 409 . . . . 5  |-  ( ( F  e.  ( J
Homeo K )  /\  A  C_  X )  ->  (
( int `  K
) `  ( F " A ) )  C_  U. K )
2625, 10sseqtrrd 3167 . . . 4  |-  ( ( F  e.  ( J
Homeo K )  /\  A  C_  X )  ->  (
( int `  K
) `  ( F " A ) )  C_  ran  F )
27 funimass1 5248 . . . 4  |-  ( ( Fun  F  /\  (
( int `  K
) `  ( F " A ) )  C_  ran  F )  ->  (
( `' F "
( ( int `  K
) `  ( F " A ) ) ) 
C_  ( ( int `  J ) `  A
)  ->  ( ( int `  K ) `  ( F " A ) )  C_  ( F " ( ( int `  J
) `  A )
) ) )
2821, 26, 27syl2anc 409 . . 3  |-  ( ( F  e.  ( J
Homeo K )  /\  A  C_  X )  ->  (
( `' F "
( ( int `  K
) `  ( F " A ) ) ) 
C_  ( ( int `  J ) `  A
)  ->  ( ( int `  K ) `  ( F " A ) )  C_  ( F " ( ( int `  J
) `  A )
) ) )
2919, 28mpd 13 . 2  |-  ( ( F  e.  ( J
Homeo K )  /\  A  C_  X )  ->  (
( int `  K
) `  ( F " A ) )  C_  ( F " ( ( int `  J ) `
 A ) ) )
30 hmeocnvcn 12748 . . . 4  |-  ( F  e.  ( J Homeo K )  ->  `' F  e.  ( K  Cn  J
) )
314cnntri 12666 . . . 4  |-  ( ( `' F  e.  ( K  Cn  J )  /\  A  C_  X )  -> 
( `' `' F " ( ( int `  J
) `  A )
)  C_  ( ( int `  K ) `  ( `' `' F " A ) ) )
3230, 31sylan 281 . . 3  |-  ( ( F  e.  ( J
Homeo K )  /\  A  C_  X )  ->  ( `' `' F " ( ( int `  J ) `
 A ) ) 
C_  ( ( int `  K ) `  ( `' `' F " A ) ) )
33 imacnvcnv 5051 . . 3  |-  ( `' `' F " ( ( int `  J ) `
 A ) )  =  ( F "
( ( int `  J
) `  A )
)
34 imacnvcnv 5051 . . . 4  |-  ( `' `' F " A )  =  ( F " A )
3534fveq2i 5472 . . 3  |-  ( ( int `  K ) `
 ( `' `' F " A ) )  =  ( ( int `  K ) `  ( F " A ) )
3632, 33, 353sstr3g 3170 . 2  |-  ( ( F  e.  ( J
Homeo K )  /\  A  C_  X )  ->  ( F " ( ( int `  J ) `  A
) )  C_  (
( int `  K
) `  ( F " A ) ) )
3729, 36eqssd 3145 1  |-  ( ( F  e.  ( J
Homeo K )  /\  A  C_  X )  ->  (
( int `  K
) `  ( F " A ) )  =  ( F " (
( int `  J
) `  A )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1335    e. wcel 2128    C_ wss 3102   U.cuni 3773   `'ccnv 4586   ran crn 4588   "cima 4590   Fun wfun 5165   -1-1->wf1 5168   -onto->wfo 5169   -1-1-onto->wf1o 5170   ` cfv 5171  (class class class)co 5825   Topctop 12437   intcnt 12535    Cn ccn 12627   Homeochmeo 12742
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4080  ax-sep 4083  ax-pow 4136  ax-pr 4170  ax-un 4394  ax-setind 4497
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3774  df-iun 3852  df-br 3967  df-opab 4027  df-mpt 4028  df-id 4254  df-xp 4593  df-rel 4594  df-cnv 4595  df-co 4596  df-dm 4597  df-rn 4598  df-res 4599  df-ima 4600  df-iota 5136  df-fun 5173  df-fn 5174  df-f 5175  df-f1 5176  df-fo 5177  df-f1o 5178  df-fv 5179  df-ov 5828  df-oprab 5829  df-mpo 5830  df-1st 6089  df-2nd 6090  df-map 6596  df-top 12438  df-topon 12451  df-ntr 12538  df-cn 12630  df-hmeo 12743
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator