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Theorem hmeontr 19467
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 19458 . . . . . 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 5281 . . . . . 6  |-  ( F
" A )  C_  ran  F
4 hmeoopn.1 . . . . . . . . 9  |-  X  = 
U. J
5 eqid 2451 . . . . . . . . 9  |-  U. K  =  U. K
64, 5hmeof1o 19462 . . . . . . . 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 5749 . . . . . . 7  |-  ( F : X -1-1-onto-> U. K  ->  F : X -onto-> U. K )
9 forn 5724 . . . . . . 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 3505 . . . . 5  |-  ( ( F  e.  ( J
Homeo K )  /\  A  C_  X )  ->  ( F " A )  C_  U. K )
125cnntri 19000 . . . . 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 5741 . . . . . . 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 5758 . . . . . 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 5796 . . . 4  |-  ( ( F  e.  ( J
Homeo K )  /\  A  C_  X )  ->  (
( int `  J
) `  ( `' F " ( F " A ) ) )  =  ( ( int `  J ) `  A
) )
1913, 18sseqtrd 3493 . . 3  |-  ( ( F  e.  ( J
Homeo K )  /\  A  C_  X )  ->  ( `' F " ( ( int `  K ) `
 ( F " A ) ) ) 
C_  ( ( int `  J ) `  A
) )
20 f1ofun 5744 . . . . 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 18970 . . . . . . 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 18789 . . . . . 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 3494 . . . 4  |-  ( ( F  e.  ( J
Homeo K )  /\  A  C_  X )  ->  (
( int `  K
) `  ( F " A ) )  C_  ran  F )
27 funimass1 5592 . . . 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 19459 . . . 4  |-  ( F  e.  ( J Homeo K )  ->  `' F  e.  ( K  Cn  J
) )
314cnntri 19000 . . . 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 5404 . . 3  |-  ( `' `' F " ( ( int `  J ) `
 A ) )  =  ( F "
( ( int `  J
) `  A )
)
34 imacnvcnv 5404 . . . 4  |-  ( `' `' F " A )  =  ( F " A )
3534fveq2i 5795 . . 3  |-  ( ( int `  K ) `
 ( `' `' F " A ) )  =  ( ( int `  K ) `  ( F " A ) )
3632, 33, 353sstr3g 3497 . 2  |-  ( ( F  e.  ( J
Homeo K )  /\  A  C_  X )  ->  ( F " ( ( int `  J ) `  A
) )  C_  (
( int `  K
) `  ( F " A ) ) )
3729, 36eqssd 3474 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 1370    e. wcel 1758    C_ wss 3429   U.cuni 4192   `'ccnv 4940   ran crn 4942   "cima 4944   Fun wfun 5513   -1-1->wf1 5516   -onto->wfo 5517   -1-1-onto->wf1o 5518   ` cfv 5519  (class class class)co 6193   Topctop 18623   intcnt 18746    Cn ccn 18953   Homeochmeo 19451
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-map 7319  df-top 18628  df-topon 18631  df-ntr 18749  df-cn 18956  df-hmeo 19453
This theorem is referenced by: (None)
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