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Theorem hmeontr 20005
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 19996 . . . . . 6  |-  ( F  e.  ( J Homeo K )  ->  F  e.  ( J  Cn  K
) )
21adantr 465 . . . . 5  |-  ( ( F  e.  ( J
Homeo K )  /\  A  C_  X )  ->  F  e.  ( J  Cn  K
) )
3 imassrn 5346 . . . . . 6  |-  ( F
" A )  C_  ran  F
4 hmeoopn.1 . . . . . . . . 9  |-  X  = 
U. J
5 eqid 2467 . . . . . . . . 9  |-  U. K  =  U. K
64, 5hmeof1o 20000 . . . . . . . 8  |-  ( F  e.  ( J Homeo K )  ->  F : X
-1-1-onto-> U. K )
76adantr 465 . . . . . . 7  |-  ( ( F  e.  ( J
Homeo K )  /\  A  C_  X )  ->  F : X -1-1-onto-> U. K )
8 f1ofo 5821 . . . . . . 7  |-  ( F : X -1-1-onto-> U. K  ->  F : X -onto-> U. K )
9 forn 5796 . . . . . . 7  |-  ( F : X -onto-> U. K  ->  ran  F  =  U. K )
107, 8, 93syl 20 . . . . . 6  |-  ( ( F  e.  ( J
Homeo K )  /\  A  C_  X )  ->  ran  F  =  U. K )
113, 10syl5sseq 3552 . . . . 5  |-  ( ( F  e.  ( J
Homeo K )  /\  A  C_  X )  ->  ( F " A )  C_  U. K )
125cnntri 19538 . . . . 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 661 . . . 4  |-  ( ( F  e.  ( J
Homeo K )  /\  A  C_  X )  ->  ( `' F " ( ( int `  K ) `
 ( F " A ) ) ) 
C_  ( ( int `  J ) `  ( `' F " ( F
" A ) ) ) )
14 f1of1 5813 . . . . . . 7  |-  ( F : X -1-1-onto-> U. K  ->  F : X -1-1-> U. K )
157, 14syl 16 . . . . . 6  |-  ( ( F  e.  ( J
Homeo K )  /\  A  C_  X )  ->  F : X -1-1-> U. K )
16 f1imacnv 5830 . . . . . 6  |-  ( ( F : X -1-1-> U. K  /\  A  C_  X
)  ->  ( `' F " ( F " A ) )  =  A )
1715, 16sylancom 667 . . . . 5  |-  ( ( F  e.  ( J
Homeo K )  /\  A  C_  X )  ->  ( `' F " ( F
" A ) )  =  A )
1817fveq2d 5868 . . . 4  |-  ( ( F  e.  ( J
Homeo K )  /\  A  C_  X )  ->  (
( int `  J
) `  ( `' F " ( F " A ) ) )  =  ( ( int `  J ) `  A
) )
1913, 18sseqtrd 3540 . . 3  |-  ( ( F  e.  ( J
Homeo K )  /\  A  C_  X )  ->  ( `' F " ( ( int `  K ) `
 ( F " A ) ) ) 
C_  ( ( int `  J ) `  A
) )
20 f1ofun 5816 . . . . 5  |-  ( F : X -1-1-onto-> U. K  ->  Fun  F )
217, 20syl 16 . . . 4  |-  ( ( F  e.  ( J
Homeo K )  /\  A  C_  X )  ->  Fun  F )
22 cntop2 19508 . . . . . . 7  |-  ( F  e.  ( J  Cn  K )  ->  K  e.  Top )
232, 22syl 16 . . . . . 6  |-  ( ( F  e.  ( J
Homeo K )  /\  A  C_  X )  ->  K  e.  Top )
245ntrss3 19327 . . . . . 6  |-  ( ( K  e.  Top  /\  ( F " A ) 
C_  U. K )  -> 
( ( int `  K
) `  ( F " A ) )  C_  U. K )
2523, 11, 24syl2anc 661 . . . . 5  |-  ( ( F  e.  ( J
Homeo K )  /\  A  C_  X )  ->  (
( int `  K
) `  ( F " A ) )  C_  U. K )
2625, 10sseqtr4d 3541 . . . 4  |-  ( ( F  e.  ( J
Homeo K )  /\  A  C_  X )  ->  (
( int `  K
) `  ( F " A ) )  C_  ran  F )
27 funimass1 5659 . . . 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 661 . . 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 15 . 2  |-  ( ( F  e.  ( J
Homeo K )  /\  A  C_  X )  ->  (
( int `  K
) `  ( F " A ) )  C_  ( F " ( ( int `  J ) `
 A ) ) )
30 hmeocnvcn 19997 . . . 4  |-  ( F  e.  ( J Homeo K )  ->  `' F  e.  ( K  Cn  J
) )
314cnntri 19538 . . . 4  |-  ( ( `' F  e.  ( K  Cn  J )  /\  A  C_  X )  -> 
( `' `' F " ( ( int `  J
) `  A )
)  C_  ( ( int `  K ) `  ( `' `' F " A ) ) )
3230, 31sylan 471 . . 3  |-  ( ( F  e.  ( J
Homeo K )  /\  A  C_  X )  ->  ( `' `' F " ( ( int `  J ) `
 A ) ) 
C_  ( ( int `  K ) `  ( `' `' F " A ) ) )
33 imacnvcnv 5470 . . 3  |-  ( `' `' F " ( ( int `  J ) `
 A ) )  =  ( F "
( ( int `  J
) `  A )
)
34 imacnvcnv 5470 . . . 4  |-  ( `' `' F " A )  =  ( F " A )
3534fveq2i 5867 . . 3  |-  ( ( int `  K ) `
 ( `' `' F " A ) )  =  ( ( int `  K ) `  ( F " A ) )
3632, 33, 353sstr3g 3544 . 2  |-  ( ( F  e.  ( J
Homeo K )  /\  A  C_  X )  ->  ( F " ( ( int `  J ) `  A
) )  C_  (
( int `  K
) `  ( F " A ) ) )
3729, 36eqssd 3521 1  |-  ( ( F  e.  ( J
Homeo K )  /\  A  C_  X )  ->  (
( int `  K
) `  ( F " A ) )  =  ( F " (
( int `  J
) `  A )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767    C_ wss 3476   U.cuni 4245   `'ccnv 4998   ran crn 5000   "cima 5002   Fun wfun 5580   -1-1->wf1 5583   -onto->wfo 5584   -1-1-onto->wf1o 5585   ` cfv 5586  (class class class)co 6282   Topctop 19161   intcnt 19284    Cn ccn 19491   Homeochmeo 19989
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-map 7419  df-top 19166  df-topon 19169  df-ntr 19287  df-cn 19494  df-hmeo 19991
This theorem is referenced by: (None)
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