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Theorem hmeontr 17460
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 17451 . . . . . 6  |-  ( F  e.  ( J  Homeo  K )  ->  F  e.  ( J  Cn  K
) )
21adantr 451 . . . . 5  |-  ( ( F  e.  ( J 
Homeo  K )  /\  A  C_  X )  ->  F  e.  ( J  Cn  K
) )
3 imassrn 5025 . . . . . 6  |-  ( F
" A )  C_  ran  F
4 hmeoopn.1 . . . . . . . . 9  |-  X  = 
U. J
5 eqid 2283 . . . . . . . . 9  |-  U. K  =  U. K
64, 5hmeof1o 17455 . . . . . . . 8  |-  ( F  e.  ( J  Homeo  K )  ->  F : X
-1-1-onto-> U. K )
76adantr 451 . . . . . . 7  |-  ( ( F  e.  ( J 
Homeo  K )  /\  A  C_  X )  ->  F : X -1-1-onto-> U. K )
8 f1ofo 5479 . . . . . . 7  |-  ( F : X -1-1-onto-> U. K  ->  F : X -onto-> U. K )
9 forn 5454 . . . . . . 7  |-  ( F : X -onto-> U. K  ->  ran  F  =  U. K )
107, 8, 93syl 18 . . . . . 6  |-  ( ( F  e.  ( J 
Homeo  K )  /\  A  C_  X )  ->  ran  F  =  U. K )
113, 10syl5sseq 3226 . . . . 5  |-  ( ( F  e.  ( J 
Homeo  K )  /\  A  C_  X )  ->  ( F " A )  C_  U. K )
125cnntri 17000 . . . . 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 642 . . . 4  |-  ( ( F  e.  ( J 
Homeo  K )  /\  A  C_  X )  ->  ( `' F " ( ( int `  K ) `
 ( F " A ) ) ) 
C_  ( ( int `  J ) `  ( `' F " ( F
" A ) ) ) )
14 f1of1 5471 . . . . . . 7  |-  ( F : X -1-1-onto-> U. K  ->  F : X -1-1-> U. K )
157, 14syl 15 . . . . . 6  |-  ( ( F  e.  ( J 
Homeo  K )  /\  A  C_  X )  ->  F : X -1-1-> U. K )
16 f1imacnv 5489 . . . . . 6  |-  ( ( F : X -1-1-> U. K  /\  A  C_  X
)  ->  ( `' F " ( F " A ) )  =  A )
1715, 16sylancom 648 . . . . 5  |-  ( ( F  e.  ( J 
Homeo  K )  /\  A  C_  X )  ->  ( `' F " ( F
" A ) )  =  A )
1817fveq2d 5529 . . . 4  |-  ( ( F  e.  ( J 
Homeo  K )  /\  A  C_  X )  ->  (
( int `  J
) `  ( `' F " ( F " A ) ) )  =  ( ( int `  J ) `  A
) )
1913, 18sseqtrd 3214 . . 3  |-  ( ( F  e.  ( J 
Homeo  K )  /\  A  C_  X )  ->  ( `' F " ( ( int `  K ) `
 ( F " A ) ) ) 
C_  ( ( int `  J ) `  A
) )
20 f1ofun 5474 . . . . 5  |-  ( F : X -1-1-onto-> U. K  ->  Fun  F )
217, 20syl 15 . . . 4  |-  ( ( F  e.  ( J 
Homeo  K )  /\  A  C_  X )  ->  Fun  F )
22 cntop2 16971 . . . . . . 7  |-  ( F  e.  ( J  Cn  K )  ->  K  e.  Top )
232, 22syl 15 . . . . . 6  |-  ( ( F  e.  ( J 
Homeo  K )  /\  A  C_  X )  ->  K  e.  Top )
245ntrss3 16797 . . . . . 6  |-  ( ( K  e.  Top  /\  ( F " A ) 
C_  U. K )  -> 
( ( int `  K
) `  ( F " A ) )  C_  U. K )
2523, 11, 24syl2anc 642 . . . . 5  |-  ( ( F  e.  ( J 
Homeo  K )  /\  A  C_  X )  ->  (
( int `  K
) `  ( F " A ) )  C_  U. K )
2625, 10sseqtr4d 3215 . . . 4  |-  ( ( F  e.  ( J 
Homeo  K )  /\  A  C_  X )  ->  (
( int `  K
) `  ( F " A ) )  C_  ran  F )
27 funimass1 5325 . . . 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 642 . . 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 14 . 2  |-  ( ( F  e.  ( J 
Homeo  K )  /\  A  C_  X )  ->  (
( int `  K
) `  ( F " A ) )  C_  ( F " ( ( int `  J ) `
 A ) ) )
30 hmeocnvcn 17452 . . . 4  |-  ( F  e.  ( J  Homeo  K )  ->  `' F  e.  ( K  Cn  J
) )
314cnntri 17000 . . . 4  |-  ( ( `' F  e.  ( K  Cn  J )  /\  A  C_  X )  -> 
( `' `' F " ( ( int `  J
) `  A )
)  C_  ( ( int `  K ) `  ( `' `' F " A ) ) )
3230, 31sylan 457 . . 3  |-  ( ( F  e.  ( J 
Homeo  K )  /\  A  C_  X )  ->  ( `' `' F " ( ( int `  J ) `
 A ) ) 
C_  ( ( int `  K ) `  ( `' `' F " A ) ) )
33 imacnvcnv 5137 . . 3  |-  ( `' `' F " ( ( int `  J ) `
 A ) )  =  ( F "
( ( int `  J
) `  A )
)
34 imacnvcnv 5137 . . . 4  |-  ( `' `' F " A )  =  ( F " A )
3534fveq2i 5528 . . 3  |-  ( ( int `  K ) `
 ( `' `' F " A ) )  =  ( ( int `  K ) `  ( F " A ) )
3632, 33, 353sstr3g 3218 . 2  |-  ( ( F  e.  ( J 
Homeo  K )  /\  A  C_  X )  ->  ( F " ( ( int `  J ) `  A
) )  C_  (
( int `  K
) `  ( F " A ) ) )
3729, 36eqssd 3196 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 358    = wceq 1623    e. wcel 1684    C_ wss 3152   U.cuni 3827   `'ccnv 4688   ran crn 4690   "cima 4692   Fun wfun 5249   -1-1->wf1 5252   -onto->wfo 5253   -1-1-onto->wf1o 5254   ` cfv 5255  (class class class)co 5858   Topctop 16631   intcnt 16754    Cn ccn 16954    Homeo chmeo 17444
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-map 6774  df-top 16636  df-topon 16639  df-ntr 16757  df-cn 16957  df-hmeo 17446
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