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Theorem hmeontr 20776
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 20767 . . . . . 6  |-  ( F  e.  ( J Homeo K )  ->  F  e.  ( J  Cn  K
) )
21adantr 467 . . . . 5  |-  ( ( F  e.  ( J
Homeo K )  /\  A  C_  X )  ->  F  e.  ( J  Cn  K
) )
3 imassrn 5196 . . . . . 6  |-  ( F
" A )  C_  ran  F
4 hmeoopn.1 . . . . . . . . 9  |-  X  = 
U. J
5 eqid 2423 . . . . . . . . 9  |-  U. K  =  U. K
64, 5hmeof1o 20771 . . . . . . . 8  |-  ( F  e.  ( J Homeo K )  ->  F : X
-1-1-onto-> U. K )
76adantr 467 . . . . . . 7  |-  ( ( F  e.  ( J
Homeo K )  /\  A  C_  X )  ->  F : X -1-1-onto-> U. K )
8 f1ofo 5836 . . . . . . 7  |-  ( F : X -1-1-onto-> U. K  ->  F : X -onto-> U. K )
9 forn 5811 . . . . . . 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 3513 . . . . 5  |-  ( ( F  e.  ( J
Homeo K )  /\  A  C_  X )  ->  ( F " A )  C_  U. K )
125cnntri 20279 . . . . 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 666 . . . 4  |-  ( ( F  e.  ( J
Homeo K )  /\  A  C_  X )  ->  ( `' F " ( ( int `  K ) `
 ( F " A ) ) ) 
C_  ( ( int `  J ) `  ( `' F " ( F
" A ) ) ) )
14 f1of1 5828 . . . . . . 7  |-  ( F : X -1-1-onto-> U. K  ->  F : X -1-1-> U. K )
157, 14syl 17 . . . . . 6  |-  ( ( F  e.  ( J
Homeo K )  /\  A  C_  X )  ->  F : X -1-1-> U. K )
16 f1imacnv 5845 . . . . . 6  |-  ( ( F : X -1-1-> U. K  /\  A  C_  X
)  ->  ( `' F " ( F " A ) )  =  A )
1715, 16sylancom 672 . . . . 5  |-  ( ( F  e.  ( J
Homeo K )  /\  A  C_  X )  ->  ( `' F " ( F
" A ) )  =  A )
1817fveq2d 5883 . . . 4  |-  ( ( F  e.  ( J
Homeo K )  /\  A  C_  X )  ->  (
( int `  J
) `  ( `' F " ( F " A ) ) )  =  ( ( int `  J ) `  A
) )
1913, 18sseqtrd 3501 . . 3  |-  ( ( F  e.  ( J
Homeo K )  /\  A  C_  X )  ->  ( `' F " ( ( int `  K ) `
 ( F " A ) ) ) 
C_  ( ( int `  J ) `  A
) )
20 f1ofun 5831 . . . . 5  |-  ( F : X -1-1-onto-> U. K  ->  Fun  F )
217, 20syl 17 . . . 4  |-  ( ( F  e.  ( J
Homeo K )  /\  A  C_  X )  ->  Fun  F )
22 cntop2 20249 . . . . . . 7  |-  ( F  e.  ( J  Cn  K )  ->  K  e.  Top )
232, 22syl 17 . . . . . 6  |-  ( ( F  e.  ( J
Homeo K )  /\  A  C_  X )  ->  K  e.  Top )
245ntrss3 20067 . . . . . 6  |-  ( ( K  e.  Top  /\  ( F " A ) 
C_  U. K )  -> 
( ( int `  K
) `  ( F " A ) )  C_  U. K )
2523, 11, 24syl2anc 666 . . . . 5  |-  ( ( F  e.  ( J
Homeo K )  /\  A  C_  X )  ->  (
( int `  K
) `  ( F " A ) )  C_  U. K )
2625, 10sseqtr4d 3502 . . . 4  |-  ( ( F  e.  ( J
Homeo K )  /\  A  C_  X )  ->  (
( int `  K
) `  ( F " A ) )  C_  ran  F )
27 funimass1 5672 . . . 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 666 . . 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 20768 . . . 4  |-  ( F  e.  ( J Homeo K )  ->  `' F  e.  ( K  Cn  J
) )
314cnntri 20279 . . . 4  |-  ( ( `' F  e.  ( K  Cn  J )  /\  A  C_  X )  -> 
( `' `' F " ( ( int `  J
) `  A )
)  C_  ( ( int `  K ) `  ( `' `' F " A ) ) )
3230, 31sylan 474 . . 3  |-  ( ( F  e.  ( J
Homeo K )  /\  A  C_  X )  ->  ( `' `' F " ( ( int `  J ) `
 A ) ) 
C_  ( ( int `  K ) `  ( `' `' F " A ) ) )
33 imacnvcnv 5317 . . 3  |-  ( `' `' F " ( ( int `  J ) `
 A ) )  =  ( F "
( ( int `  J
) `  A )
)
34 imacnvcnv 5317 . . . 4  |-  ( `' `' F " A )  =  ( F " A )
3534fveq2i 5882 . . 3  |-  ( ( int `  K ) `
 ( `' `' F " A ) )  =  ( ( int `  K ) `  ( F " A ) )
3632, 33, 353sstr3g 3505 . 2  |-  ( ( F  e.  ( J
Homeo K )  /\  A  C_  X )  ->  ( F " ( ( int `  J ) `  A
) )  C_  (
( int `  K
) `  ( F " A ) ) )
3729, 36eqssd 3482 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 371    = wceq 1438    e. wcel 1869    C_ wss 3437   U.cuni 4217   `'ccnv 4850   ran crn 4852   "cima 4854   Fun wfun 5593   -1-1->wf1 5596   -onto->wfo 5597   -1-1-onto->wf1o 5598   ` cfv 5599  (class class class)co 6303   Topctop 19909   intcnt 20024    Cn ccn 20232   Homeochmeo 20760
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-rep 4534  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-reu 2783  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-op 4004  df-uni 4218  df-iun 4299  df-br 4422  df-opab 4481  df-mpt 4482  df-id 4766  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-ov 6306  df-oprab 6307  df-mpt2 6308  df-map 7480  df-top 19913  df-topon 19915  df-ntr 20027  df-cn 20235  df-hmeo 20762
This theorem is referenced by: (None)
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