Theorem hmeontr 15414

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 15406	. . . . . 6 |- ( F e. ( J Homeo K ) -> F e. ( J Cn K ) )
2	1	adantr 276	. . . . 5 |- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> F e. ( J Cn K ) )
3		imassrn 5137	. . . . . 6 |- ( F " A ) C_ ran F
4		hmeoopn.1	. . . . . . . . 9 |- X = U. J
5		eqid 2238	. . . . . . . . 9 |- U. K = U. K
6	4, 5	hmeof1o 15410	. . . . . . . 8 |- ( F e. ( J Homeo K ) -> F : X -1-1-onto-> U. K )
7	6	adantr 276	. . . . . . 7 |- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> F : X -1-1-onto-> U. K )
8		f1ofo 5646	. . . . . . 7 |- ( F : X -1-1-onto-> U. K -> F : X -onto-> U. K )
9		forn 5618	. . . . . . 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 3298	. . . . 5 |- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> ( F " A ) C_ U. K )
12	5	cnntri 15325	. . . . 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 415	. . . 4 |- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> ( `' F " ( ( int ` K ) ` ( F " A ) ) ) C_ ( ( int ` J ) ` ( `' F " ( F " A ) ) ) )
14		f1of1 5638	. . . . . . 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 5656	. . . . . 6 |- ( ( F : X -1-1-> U. K /\ A C_ X ) -> ( `' F " ( F " A ) ) = A )
17	15, 16	sylancom 424	. . . . 5 |- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> ( `' F " ( F " A ) ) = A )
18	17	fveq2d 5699	. . . 4 |- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> ( ( int ` J ) ` ( `' F " ( F " A ) ) ) = ( ( int ` J ) ` A ) )
19	13, 18	sseqtrd 3286	. . 3 |- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> ( `' F " ( ( int ` K ) ` ( F " A ) ) ) C_ ( ( int ` J ) ` A ) )
20		f1ofun 5641	. . . . 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 15303	. . . . . . 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 15224	. . . . . 6 |- ( ( K e. Top /\ ( F " A ) C_ U. K ) -> ( ( int ` K ) ` ( F " A ) ) C_ U. K )
25	23, 11, 24	syl2anc 415	. . . . 5 |- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> ( ( int ` K ) ` ( F " A ) ) C_ U. K )
26	25, 10	sseqtrrd 3287	. . . 4 |- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> ( ( int ` K ) ` ( F " A ) ) C_ ran F )
27		funimass1 5458	. . . 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 415	. . 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 15407	. . . 4 |- ( F e. ( J Homeo K ) -> `' F e. ( K Cn J ) )
31	4	cnntri 15325	. . . 4 |- ( ( `' F e. ( K Cn J ) /\ A C_ X ) -> ( `' `' F " ( ( int ` J ) ` A ) ) C_ ( ( int ` K ) ` ( `' `' F " A ) ) )
32	30, 31	sylan 283	. . 3 |- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> ( `' `' F " ( ( int ` J ) ` A ) ) C_ ( ( int ` K ) ` ( `' `' F " A ) ) )
33		imacnvcnv 5252	. . 3 |- ( `' `' F " ( ( int ` J ) ` A ) ) = ( F " ( ( int ` J ) ` A ) )
34		imacnvcnv 5252	. . . 4 |- ( `' `' F " A ) = ( F " A )
35	34	fveq2i 5698	. . 3 |- ( ( int ` K ) ` ( `' `' F " A ) ) = ( ( int ` K ) ` ( F " A ) )
36	32, 33, 35	3sstr3g 3290	. 2 |- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> ( F " ( ( int ` J ) ` A ) ) C_ ( ( int ` K ) ` ( F " A ) ) )
37	29, 36	eqssd 3265	1 |- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> ( ( int ` K ) ` ( F " A ) ) = ( F " ( ( int ` J ) ` A ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   C_ wss 3220   U.cuni 3935   `'ccnv 4773   ran crn 4775   "cima 4777   Fun wfun 5371   -1-1->wf1 5374   -onto->wfo 5375   -1-1-onto->wf1o 5376   ` cfv 5377  (class class class)co 6085   Topctop 15098   intcnt 15194   Cn ccn 15286   Homeochmeo 15401

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-map 6924  df-top 15099  df-topon 15112  df-ntr 15197  df-cn 15289  df-hmeo 15402

This theorem is used by: (None)