Theorem hmeontr 13107

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 13099	. . . . . 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 4964	. . . . . 6 |- ( F " A ) C_ ran F
4		hmeoopn.1	. . . . . . . . 9 |- X = U. J
5		eqid 2170	. . . . . . . . 9 |- U. K = U. K
6	4, 5	hmeof1o 13103	. . . . . . . 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 5449	. . . . . . 7 |- ( F : X -1-1-onto-> U. K -> F : X -onto-> U. K )
9		forn 5423	. . . . . . 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 3197	. . . . 5 |- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> ( F " A ) C_ U. K )
12	5	cnntri 13018	. . . . 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 5441	. . . . . . 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 5459	. . . . . 6 |- ( ( F : X -1-1-> U. K /\ A C_ X ) -> ( `' F " ( F " A ) ) = A )
17	15, 16	sylancom 418	. . . . 5 |- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> ( `' F " ( F " A ) ) = A )
18	17	fveq2d 5500	. . . 4 |- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> ( ( int ` J ) ` ( `' F " ( F " A ) ) ) = ( ( int ` J ) ` A ) )
19	13, 18	sseqtrd 3185	. . 3 |- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> ( `' F " ( ( int ` K ) ` ( F " A ) ) ) C_ ( ( int ` J ) ` A ) )
20		f1ofun 5444	. . . . 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 12996	. . . . . . 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 12917	. . . . . 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 3186	. . . 4 |- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> ( ( int ` K ) ` ( F " A ) ) C_ ran F )
27		funimass1 5275	. . . 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 13100	. . . 4 |- ( F e. ( J Homeo K ) -> `' F e. ( K Cn J ) )
31	4	cnntri 13018	. . . 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 5075	. . 3 |- ( `' `' F " ( ( int ` J ) ` A ) ) = ( F " ( ( int ` J ) ` A ) )
34		imacnvcnv 5075	. . . 4 |- ( `' `' F " A ) = ( F " A )
35	34	fveq2i 5499	. . 3 |- ( ( int ` K ) ` ( `' `' F " A ) ) = ( ( int ` K ) ` ( F " A ) )
36	32, 33, 35	3sstr3g 3189	. 2 |- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> ( F " ( ( int ` J ) ` A ) ) C_ ( ( int ` K ) ` ( F " A ) ) )
37	29, 36	eqssd 3164	1 |- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> ( ( int ` K ) ` ( F " A ) ) = ( F " ( ( int ` J ) ` A ) ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1348   e. wcel 2141   C_ wss 3121   U.cuni 3796   `'ccnv 4610   ran crn 4612   "cima 4614   Fun wfun 5192   -1-1->wf1 5195   -onto->wfo 5196   -1-1-onto->wf1o 5197   ` cfv 5198  (class class class)co 5853   Topctop 12789   intcnt 12887   Cn ccn 12979   Homeochmeo 13094

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-map 6628  df-top 12790  df-topon 12803  df-ntr 12890  df-cn 12982  df-hmeo 13095

This theorem is referenced by: (None)