Theorem cnntri 21075

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 21044	. . 3 |- ( F e. ( J Cn K ) -> J e. Top )
2	1	adantr 481	. 2 |- ( ( F e. ( J Cn K ) /\ S C_ Y ) -> J e. Top )
3		cnvimass 5485	. . 3 |- ( `' F " S ) C_ dom F
4		eqid 2622	. . . . . 6 |- U. J = U. J
5		cncls2i.1	. . . . . 6 |- Y = U. K
6	4, 5	cnf 21050	. . . . 5 |- ( F e. ( J Cn K ) -> F : U. J --> Y )
7		fdm 6051	. . . . 5 |- ( F : U. J --> Y -> dom F = U. J )
8	6, 7	syl 17	. . . 4 |- ( F e. ( J Cn K ) -> dom F = U. J )
9	8	adantr 481	. . 3 |- ( ( F e. ( J Cn K ) /\ S C_ Y ) -> dom F = U. J )
10	3, 9	syl5sseq 3653	. 2 |- ( ( F e. ( J Cn K ) /\ S C_ Y ) -> ( `' F " S ) C_ U. J )
11		cntop2 21045	. . . 4 |- ( F e. ( J Cn K ) -> K e. Top )
12	5	ntropn 20853	. . . 4 |- ( ( K e. Top /\ S C_ Y ) -> ( ( int ` K ) ` S ) e. K )
13	11, 12	sylan 488	. . 3 |- ( ( F e. ( J Cn K ) /\ S C_ Y ) -> ( ( int ` K ) ` S ) e. K )
14		cnima 21069	. . 3 |- ( ( F e. ( J Cn K ) /\ ( ( int ` K ) ` S ) e. K ) -> ( `' F " ( ( int ` K ) ` S ) ) e. J )
15	13, 14	syldan 487	. 2 |- ( ( F e. ( J Cn K ) /\ S C_ Y ) -> ( `' F " ( ( int ` K ) ` S ) ) e. J )
16	5	ntrss2 20861	. . . 4 |- ( ( K e. Top /\ S C_ Y ) -> ( ( int ` K ) ` S ) C_ S )
17	11, 16	sylan 488	. . 3 |- ( ( F e. ( J Cn K ) /\ S C_ Y ) -> ( ( int ` K ) ` S ) C_ S )
18		imass2 5501	. . 3 |- ( ( ( int ` K ) ` S ) C_ S -> ( `' F " ( ( int ` K ) ` S ) ) C_ ( `' F " S ) )
19	17, 18	syl 17	. 2 |- ( ( F e. ( J Cn K ) /\ S C_ Y ) -> ( `' F " ( ( int ` K ) ` S ) ) C_ ( `' F " S ) )
20	4	ssntr 20862	. 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 ) ) )
21	2, 10, 15, 19, 20	syl22anc 1327	1 |- ( ( F e. ( J Cn K ) /\ S C_ Y ) -> ( `' F " ( ( int ` K ) ` S ) ) C_ ( ( int ` J ) ` ( `' F " S ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   C_ wss 3574   U.cuni 4436   `'ccnv 5113   dom cdm 5114   "cima 5117   -->wf 5884   ` cfv 5888  (class class class)co 6650   Topctop 20698   intcnt 20821   Cn ccn 21028

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-top 20699  df-topon 20716  df-ntr 20824  df-cn 21031

This theorem is referenced by:  cnntr  21079  hmeontr  21572