Theorem cnntri 12174

Description: Property of the preimage of an interior. (Contributed by Mario Carneiro, 25-Aug-2015.)

Hypothesis

Ref	Expression
cncls2i.1	|- Y = U. K

Assertion

Ref	Expression
cnntri	|- ( ( F e. ( J Cn K ) /\ S C_ Y ) -> ( `' F " ( ( int ` K ) ` S ) ) C_ ( ( int ` J ) ` ( `' F " S ) ) )

Proof of Theorem cnntri

Step	Hyp	Ref	Expression
1		cntop1 12151	. . 3 |- ( F e. ( J Cn K ) -> J e. Top )
2	1	adantr 272	. 2 |- ( ( F e. ( J Cn K ) /\ S C_ Y ) -> J e. Top )
3		cnvimass 4838	. . 3 |- ( `' F " S ) C_ dom F
4		eqid 2100	. . . . . 6 |- U. J = U. J
5		cncls2i.1	. . . . . 6 |- Y = U. K
6	4, 5	cnf 12154	. . . . 5 |- ( F e. ( J Cn K ) -> F : U. J --> Y )
7	6	fdmd 5215	. . . 4 |- ( F e. ( J Cn K ) -> dom F = U. J )
8	7	adantr 272	. . 3 |- ( ( F e. ( J Cn K ) /\ S C_ Y ) -> dom F = U. J )
9	3, 8	syl5sseq 3097	. 2 |- ( ( F e. ( J Cn K ) /\ S C_ Y ) -> ( `' F " S ) C_ U. J )
10		cntop2 12152	. . . 4 |- ( F e. ( J Cn K ) -> K e. Top )
11	5	ntropn 12068	. . . 4 |- ( ( K e. Top /\ S C_ Y ) -> ( ( int ` K ) ` S ) e. K )
12	10, 11	sylan 279	. . 3 |- ( ( F e. ( J Cn K ) /\ S C_ Y ) -> ( ( int ` K ) ` S ) e. K )
13		cnima 12170	. . 3 |- ( ( F e. ( J Cn K ) /\ ( ( int ` K ) ` S ) e. K ) -> ( `' F " ( ( int ` K ) ` S ) ) e. J )
14	12, 13	syldan 278	. 2 |- ( ( F e. ( J Cn K ) /\ S C_ Y ) -> ( `' F " ( ( int ` K ) ` S ) ) e. J )
15	5	ntrss2 12072	. . . 4 |- ( ( K e. Top /\ S C_ Y ) -> ( ( int ` K ) ` S ) C_ S )
16	10, 15	sylan 279	. . 3 |- ( ( F e. ( J Cn K ) /\ S C_ Y ) -> ( ( int ` K ) ` S ) C_ S )
17		imass2 4851	. . 3 |- ( ( ( int ` K ) ` S ) C_ S -> ( `' F " ( ( int ` K ) ` S ) ) C_ ( `' F " S ) )
18	16, 17	syl 14	. 2 |- ( ( F e. ( J Cn K ) /\ S C_ Y ) -> ( `' F " ( ( int ` K ) ` S ) ) C_ ( `' F " S ) )
19	4	ssntr 12073	. 2 |- ( ( ( J e. Top /\ ( `' F " S ) C_ U. J ) /\ ( ( `' F " ( ( int ` K ) ` S ) ) e. J /\ ( `' F " ( ( int ` K ) ` S ) ) C_ ( `' F " S ) ) ) -> ( `' F " ( ( int ` K ) ` S ) ) C_ ( ( int ` J ) ` ( `' F " S ) ) )
20	2, 9, 14, 18, 19	syl22anc 1185	1 |- ( ( F e. ( J Cn K ) /\ S C_ Y ) -> ( `' F " ( ( int ` K ) ` S ) ) C_ ( ( int ` J ) ` ( `' F " S ) ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1299   e. wcel 1448   C_ wss 3021   U.cuni 3683   `'ccnv 4476   dom cdm 4477   "cima 4480   ` cfv 5059  (class class class)co 5706   Topctop 11946   intcnt 12044   Cn ccn 12136

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-coll 3983  ax-sep 3986  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390

This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-ral 2380  df-rex 2381  df-reu 2382  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-id 4153  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-ov 5709  df-oprab 5710  df-mpo 5711  df-1st 5969  df-2nd 5970  df-map 6474  df-top 11947  df-topon 11960  df-ntr 12047  df-cn 12139

This theorem is referenced by:  cnntr  12175