Theorem hmeontr 12963

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 12955	. . . . . 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 4957	. . . . . 6 |- ( F " A ) C_ ran F
4		hmeoopn.1	. . . . . . . . 9 |- X = U. J
5		eqid 2165	. . . . . . . . 9 |- U. K = U. K
6	4, 5	hmeof1o 12959	. . . . . . . 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 5439	. . . . . . 7 |- ( F : X -1-1-onto-> U. K -> F : X -onto-> U. K )
9		forn 5413	. . . . . . 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 3192	. . . . 5 |- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> ( F " A ) C_ U. K )
12	5	cnntri 12874	. . . . 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 409	. . . 4 |- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> ( `' F " ( ( int ` K ) ` ( F " A ) ) ) C_ ( ( int ` J ) ` ( `' F " ( F " A ) ) ) )
14		f1of1 5431	. . . . . . 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 5449	. . . . . 6 |- ( ( F : X -1-1-> U. K /\ A C_ X ) -> ( `' F " ( F " A ) ) = A )
17	15, 16	sylancom 417	. . . . 5 |- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> ( `' F " ( F " A ) ) = A )
18	17	fveq2d 5490	. . . 4 |- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> ( ( int ` J ) ` ( `' F " ( F " A ) ) ) = ( ( int ` J ) ` A ) )
19	13, 18	sseqtrd 3180	. . 3 |- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> ( `' F " ( ( int ` K ) ` ( F " A ) ) ) C_ ( ( int ` J ) ` A ) )
20		f1ofun 5434	. . . . 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 12852	. . . . . . 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 12773	. . . . . 6 |- ( ( K e. Top /\ ( F " A ) C_ U. K ) -> ( ( int ` K ) ` ( F " A ) ) C_ U. K )
25	23, 11, 24	syl2anc 409	. . . . 5 |- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> ( ( int ` K ) ` ( F " A ) ) C_ U. K )
26	25, 10	sseqtrrd 3181	. . . 4 |- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> ( ( int ` K ) ` ( F " A ) ) C_ ran F )
27		funimass1 5265	. . . 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 409	. . 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 12956	. . . 4 |- ( F e. ( J Homeo K ) -> `' F e. ( K Cn J ) )
31	4	cnntri 12874	. . . 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 5068	. . 3 |- ( `' `' F " ( ( int ` J ) ` A ) ) = ( F " ( ( int ` J ) ` A ) )
34		imacnvcnv 5068	. . . 4 |- ( `' `' F " A ) = ( F " A )
35	34	fveq2i 5489	. . 3 |- ( ( int ` K ) ` ( `' `' F " A ) ) = ( ( int ` K ) ` ( F " A ) )
36	32, 33, 35	3sstr3g 3184	. 2 |- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> ( F " ( ( int ` J ) ` A ) ) C_ ( ( int ` K ) ` ( F " A ) ) )
37	29, 36	eqssd 3159	1 |- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> ( ( int ` K ) ` ( F " A ) ) = ( F " ( ( int ` J ) ` A ) ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1343   e. wcel 2136   C_ wss 3116   U.cuni 3789   `'ccnv 4603   ran crn 4605   "cima 4607   Fun wfun 5182   -1-1->wf1 5185   -onto->wfo 5186   -1-1-onto->wf1o 5187   ` cfv 5188  (class class class)co 5842   Topctop 12645   intcnt 12743   Cn ccn 12835   Homeochmeo 12950

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-map 6616  df-top 12646  df-topon 12659  df-ntr 12746  df-cn 12838  df-hmeo 12951

This theorem is referenced by: (None)