Theorem hmeontr 12482

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

Step	Hyp	Ref	Expression
1		hmeocn 12474	. . . . . 6 |- ( F e. ( J Homeo K ) -> F e. ( J Cn K ) )
2	1	adantr 274	. . . . 5 |- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> F e. ( J Cn K ) )
3		imassrn 4892	. . . . . 6 |- ( F " A ) C_ ran F
4		hmeoopn.1	. . . . . . . . 9 |- X = U. J
5		eqid 2139	. . . . . . . . 9 |- U. K = U. K
6	4, 5	hmeof1o 12478	. . . . . . . 8 |- ( F e. ( J Homeo K ) -> F : X -1-1-onto-> U. K )
7	6	adantr 274	. . . . . . 7 |- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> F : X -1-1-onto-> U. K )
8		f1ofo 5374	. . . . . . 7 |- ( F : X -1-1-onto-> U. K -> F : X -onto-> U. K )
9		forn 5348	. . . . . . 7 |- ( F : X -onto-> U. K -> ran F = U. K )
10	7, 8, 9	3syl 17	. . . . . 6 |- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> ran F = U. K )
11	3, 10	sseqtrid 3147	. . . . 5 |- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> ( F " A ) C_ U. K )
12	5	cnntri 12393	. . . . 5 |- ( ( F e. ( J Cn K ) /\ ( F " A ) C_ U. K ) -> ( `' F " ( ( int ` K ) ` ( F " A ) ) ) C_ ( ( int ` J ) ` ( `' F " ( F " A ) ) ) )
13	2, 11, 12	syl2anc 408	. . . 4 |- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> ( `' F " ( ( int ` K ) ` ( F " A ) ) ) C_ ( ( int ` J ) ` ( `' F " ( F " A ) ) ) )
14		f1of1 5366	. . . . . . 7 |- ( F : X -1-1-onto-> U. K -> F : X -1-1-> U. K )
15	7, 14	syl 14	. . . . . 6 |- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> F : X -1-1-> U. K )
16		f1imacnv 5384	. . . . . 6 |- ( ( F : X -1-1-> U. K /\ A C_ X ) -> ( `' F " ( F " A ) ) = A )
17	15, 16	sylancom 416	. . . . 5 |- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> ( `' F " ( F " A ) ) = A )
18	17	fveq2d 5425	. . . 4 |- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> ( ( int ` J ) ` ( `' F " ( F " A ) ) ) = ( ( int ` J ) ` A ) )
19	13, 18	sseqtrd 3135	. . 3 |- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> ( `' F " ( ( int ` K ) ` ( F " A ) ) ) C_ ( ( int ` J ) ` A ) )
20		f1ofun 5369	. . . . 5 |- ( F : X -1-1-onto-> U. K -> Fun F )
21	7, 20	syl 14	. . . 4 |- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> Fun F )
22		cntop2 12371	. . . . . . 7 |- ( F e. ( J Cn K ) -> K e. Top )
23	2, 22	syl 14	. . . . . 6 |- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> K e. Top )
24	5	ntrss3 12292	. . . . . 6 |- ( ( K e. Top /\ ( F " A ) C_ U. K ) -> ( ( int ` K ) ` ( F " A ) ) C_ U. K )
25	23, 11, 24	syl2anc 408	. . . . 5 |- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> ( ( int ` K ) ` ( F " A ) ) C_ U. K )
26	25, 10	sseqtrrd 3136	. . . 4 |- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> ( ( int ` K ) ` ( F " A ) ) C_ ran F )
27		funimass1 5200	. . . 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 ) ) ) )
28	21, 26, 27	syl2anc 408	. . 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 ) ) ) )
29	19, 28	mpd 13	. 2 |- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> ( ( int ` K ) ` ( F " A ) ) C_ ( F " ( ( int ` J ) ` A ) ) )
30		hmeocnvcn 12475	. . . 4 |- ( F e. ( J Homeo K ) -> `' F e. ( K Cn J ) )
31	4	cnntri 12393	. . . 4 |- ( ( `' F e. ( K Cn J ) /\ A C_ X ) -> ( `' `' F " ( ( int ` J ) ` A ) ) C_ ( ( int ` K ) ` ( `' `' F " A ) ) )
32	30, 31	sylan 281	. . 3 |- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> ( `' `' F " ( ( int ` J ) ` A ) ) C_ ( ( int ` K ) ` ( `' `' F " A ) ) )
33		imacnvcnv 5003	. . 3 |- ( `' `' F " ( ( int ` J ) ` A ) ) = ( F " ( ( int ` J ) ` A ) )
34		imacnvcnv 5003	. . . 4 |- ( `' `' F " A ) = ( F " A )
35	34	fveq2i 5424	. . 3 |- ( ( int ` K ) ` ( `' `' F " A ) ) = ( ( int ` K ) ` ( F " A ) )
36	32, 33, 35	3sstr3g 3139	. 2 |- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> ( F " ( ( int ` J ) ` A ) ) C_ ( ( int ` K ) ` ( F " A ) ) )
37	29, 36	eqssd 3114	1 |- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> ( ( int ` K ) ` ( F " A ) ) = ( F " ( ( int ` J ) ` A ) ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1331   e. wcel 1480   C_ wss 3071   U.cuni 3736   `'ccnv 4538   ran crn 4540   "cima 4542   Fun wfun 5117   -1-1->wf1 5120   -onto->wfo 5121   -1-1-onto->wf1o 5122   ` cfv 5123  (class class class)co 5774   Topctop 12164   intcnt 12262   Cn ccn 12354   Homeochmeo 12469

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-map 6544  df-top 12165  df-topon 12178  df-ntr 12265  df-cn 12357  df-hmeo 12470

This theorem is referenced by: (None)