Theorem ntrss 20859

Description: Subset relationship for interior. (Contributed by NM, 3-Oct-2007.)

Hypothesis

Ref	Expression
clscld.1	|- X = U. J

Assertion

Ref	Expression
ntrss	|- ( ( J e. Top /\ S C_ X /\ T C_ S ) -> ( ( int ` J ) ` T ) C_ ( ( int ` J ) ` S ) )

Proof of Theorem ntrss

Step	Hyp	Ref	Expression
1		sscon 3744	. . . . . . 7 |- ( T C_ S -> ( X \ S ) C_ ( X \ T ) )
2	1	adantl 482	. . . . . 6 |- ( ( S C_ X /\ T C_ S ) -> ( X \ S ) C_ ( X \ T ) )
3		difss 3737	. . . . . 6 |- ( X \ T ) C_ X
4	2, 3	jctil 560	. . . . 5 |- ( ( S C_ X /\ T C_ S ) -> ( ( X \ T ) C_ X /\ ( X \ S ) C_ ( X \ T ) ) )
5		clscld.1	. . . . . . 7 |- X = U. J
6	5	clsss 20858	. . . . . 6 |- ( ( J e. Top /\ ( X \ T ) C_ X /\ ( X \ S ) C_ ( X \ T ) ) -> ( ( cls ` J ) ` ( X \ S ) ) C_ ( ( cls ` J ) ` ( X \ T ) ) )
7	6	3expb 1266	. . . . 5 |- ( ( J e. Top /\ ( ( X \ T ) C_ X /\ ( X \ S ) C_ ( X \ T ) ) ) -> ( ( cls ` J ) ` ( X \ S ) ) C_ ( ( cls ` J ) ` ( X \ T ) ) )
8	4, 7	sylan2 491	. . . 4 |- ( ( J e. Top /\ ( S C_ X /\ T C_ S ) ) -> ( ( cls ` J ) ` ( X \ S ) ) C_ ( ( cls ` J ) ` ( X \ T ) ) )
9	8	sscond 3747	. . 3 |- ( ( J e. Top /\ ( S C_ X /\ T C_ S ) ) -> ( X \ ( ( cls ` J ) ` ( X \ T ) ) ) C_ ( X \ ( ( cls ` J ) ` ( X \ S ) ) ) )
10		sstr2 3610	. . . . 5 |- ( T C_ S -> ( S C_ X -> T C_ X ) )
11	10	impcom 446	. . . 4 |- ( ( S C_ X /\ T C_ S ) -> T C_ X )
12	5	ntrval2 20855	. . . 4 |- ( ( J e. Top /\ T C_ X ) -> ( ( int ` J ) ` T ) = ( X \ ( ( cls ` J ) ` ( X \ T ) ) ) )
13	11, 12	sylan2 491	. . 3 |- ( ( J e. Top /\ ( S C_ X /\ T C_ S ) ) -> ( ( int ` J ) ` T ) = ( X \ ( ( cls ` J ) ` ( X \ T ) ) ) )
14	5	ntrval2 20855	. . . 4 |- ( ( J e. Top /\ S C_ X ) -> ( ( int ` J ) ` S ) = ( X \ ( ( cls ` J ) ` ( X \ S ) ) ) )
15	14	adantrr 753	. . 3 |- ( ( J e. Top /\ ( S C_ X /\ T C_ S ) ) -> ( ( int ` J ) ` S ) = ( X \ ( ( cls ` J ) ` ( X \ S ) ) ) )
16	9, 13, 15	3sstr4d 3648	. 2 |- ( ( J e. Top /\ ( S C_ X /\ T C_ S ) ) -> ( ( int ` J ) ` T ) C_ ( ( int ` J ) ` S ) )
17	16	3impb 1260	1 |- ( ( J e. Top /\ S C_ X /\ T C_ S ) -> ( ( int ` J ) ` T ) C_ ( ( int ` J ) ` S ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   \ cdif 3571   C_ wss 3574   U.cuni 4436   ` cfv 5888   Topctop 20698   intcnt 20821   clsccl 20822

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-int 4476  df-iun 4522  df-iin 4523  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-top 20699  df-cld 20823  df-ntr 20824  df-cls 20825

This theorem is referenced by:  ntrin  20865  ntrcls0  20880  dvreslem  23673  dvres2lem  23674  dvaddbr  23701  dvmulbr  23702  dvcnvrelem2  23781  ntruni  32322  cldregopn  32326  limciccioolb  39853  limcicciooub  39869  cncfiooicclem1  40106