Theorem hmeontr 20560

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 20551	. . . . . 6 |- ( F e. ( J Homeo K ) -> F e. ( J Cn K ) )
2	1	adantr 463	. . . . 5 |- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> F e. ( J Cn K ) )
3		imassrn 5167	. . . . . 6 |- ( F " A ) C_ ran F
4		hmeoopn.1	. . . . . . . . 9 |- X = U. J
5		eqid 2402	. . . . . . . . 9 |- U. K = U. K
6	4, 5	hmeof1o 20555	. . . . . . . 8 |- ( F e. ( J Homeo K ) -> F : X -1-1-onto-> U. K )
7	6	adantr 463	. . . . . . 7 |- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> F : X -1-1-onto-> U. K )
8		f1ofo 5805	. . . . . . 7 |- ( F : X -1-1-onto-> U. K -> F : X -onto-> U. K )
9		forn 5780	. . . . . . 7 |- ( F : X -onto-> U. K -> ran F = U. K )
10	7, 8, 9	3syl 20	. . . . . 6 |- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> ran F = U. K )
11	3, 10	syl5sseq 3489	. . . . 5 |- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> ( F " A ) C_ U. K )
12	5	cnntri 20063	. . . . 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 659	. . . 4 |- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> ( `' F " ( ( int ` K ) ` ( F " A ) ) ) C_ ( ( int ` J ) ` ( `' F " ( F " A ) ) ) )
14		f1of1 5797	. . . . . . 7 |- ( F : X -1-1-onto-> U. K -> F : X -1-1-> U. K )
15	7, 14	syl 17	. . . . . 6 |- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> F : X -1-1-> U. K )
16		f1imacnv 5814	. . . . . 6 |- ( ( F : X -1-1-> U. K /\ A C_ X ) -> ( `' F " ( F " A ) ) = A )
17	15, 16	sylancom 665	. . . . 5 |- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> ( `' F " ( F " A ) ) = A )
18	17	fveq2d 5852	. . . 4 |- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> ( ( int ` J ) ` ( `' F " ( F " A ) ) ) = ( ( int ` J ) ` A ) )
19	13, 18	sseqtrd 3477	. . 3 |- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> ( `' F " ( ( int ` K ) ` ( F " A ) ) ) C_ ( ( int ` J ) ` A ) )
20		f1ofun 5800	. . . . 5 |- ( F : X -1-1-onto-> U. K -> Fun F )
21	7, 20	syl 17	. . . 4 |- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> Fun F )
22		cntop2 20033	. . . . . . 7 |- ( F e. ( J Cn K ) -> K e. Top )
23	2, 22	syl 17	. . . . . 6 |- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> K e. Top )
24	5	ntrss3 19851	. . . . . 6 |- ( ( K e. Top /\ ( F " A ) C_ U. K ) -> ( ( int ` K ) ` ( F " A ) ) C_ U. K )
25	23, 11, 24	syl2anc 659	. . . . 5 |- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> ( ( int ` K ) ` ( F " A ) ) C_ U. K )
26	25, 10	sseqtr4d 3478	. . . 4 |- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> ( ( int ` K ) ` ( F " A ) ) C_ ran F )
27		funimass1 5641	. . . 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 659	. . 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 15	. 2 |- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> ( ( int ` K ) ` ( F " A ) ) C_ ( F " ( ( int ` J ) ` A ) ) )
30		hmeocnvcn 20552	. . . 4 |- ( F e. ( J Homeo K ) -> `' F e. ( K Cn J ) )
31	4	cnntri 20063	. . . 4 |- ( ( `' F e. ( K Cn J ) /\ A C_ X ) -> ( `' `' F " ( ( int ` J ) ` A ) ) C_ ( ( int ` K ) ` ( `' `' F " A ) ) )
32	30, 31	sylan 469	. . 3 |- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> ( `' `' F " ( ( int ` J ) ` A ) ) C_ ( ( int ` K ) ` ( `' `' F " A ) ) )
33		imacnvcnv 5287	. . 3 |- ( `' `' F " ( ( int ` J ) ` A ) ) = ( F " ( ( int ` J ) ` A ) )
34		imacnvcnv 5287	. . . 4 |- ( `' `' F " A ) = ( F " A )
35	34	fveq2i 5851	. . 3 |- ( ( int ` K ) ` ( `' `' F " A ) ) = ( ( int ` K ) ` ( F " A ) )
36	32, 33, 35	3sstr3g 3481	. 2 |- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> ( F " ( ( int ` J ) ` A ) ) C_ ( ( int ` K ) ` ( F " A ) ) )
37	29, 36	eqssd 3458	1 |- ( ( F e. ( J Homeo K ) /\ A C_ X ) -> ( ( int ` K ) ` ( F " A ) ) = ( F " ( ( int ` J ) ` A ) ) )

Syntax hints:   -> wi 4   /\ wa 367   = wceq 1405   e. wcel 1842   C_ wss 3413   U.cuni 4190   `'ccnv 4821   ran crn 4823   "cima 4825   Fun wfun 5562   -1-1->wf1 5565   -onto->wfo 5566   -1-1-onto->wf1o 5567   ` cfv 5568  (class class class)co 6277   Topctop 19684   intcnt 19808   Cn ccn 20016   Homeochmeo 20544

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-map 7458  df-top 19689  df-topon 19692  df-ntr 19811  df-cn 20019  df-hmeo 20546

This theorem is referenced by: (None)