Theorem ntrss 12490

Description: Subset relationship for interior. (Contributed by NM, 3-Oct-2007.) (Revised by Jim Kingdon, 11-Mar-2023.)

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		simp3 984	. . 3 |- ( ( J e. Top /\ S C_ X /\ T C_ S ) -> T C_ S )
2		sspwb 4176	. . . . 5 |- ( T C_ S <-> ~P T C_ ~P S )
3		sslin 3333	. . . . 5 |- ( ~P T C_ ~P S -> ( J i^i ~P T ) C_ ( J i^i ~P S ) )
4	2, 3	sylbi 120	. . . 4 |- ( T C_ S -> ( J i^i ~P T ) C_ ( J i^i ~P S ) )
5	4	unissd 3796	. . 3 |- ( T C_ S -> U. ( J i^i ~P T ) C_ U. ( J i^i ~P S ) )
6	1, 5	syl 14	. 2 |- ( ( J e. Top /\ S C_ X /\ T C_ S ) -> U. ( J i^i ~P T ) C_ U. ( J i^i ~P S ) )
7		simp1 982	. . 3 |- ( ( J e. Top /\ S C_ X /\ T C_ S ) -> J e. Top )
8		simp2 983	. . . 4 |- ( ( J e. Top /\ S C_ X /\ T C_ S ) -> S C_ X )
9	1, 8	sstrd 3138	. . 3 |- ( ( J e. Top /\ S C_ X /\ T C_ S ) -> T C_ X )
10		clscld.1	. . . 4 |- X = U. J
11	10	ntrval 12481	. . 3 |- ( ( J e. Top /\ T C_ X ) -> ( ( int ` J ) ` T ) = U. ( J i^i ~P T ) )
12	7, 9, 11	syl2anc 409	. 2 |- ( ( J e. Top /\ S C_ X /\ T C_ S ) -> ( ( int ` J ) ` T ) = U. ( J i^i ~P T ) )
13	10	ntrval 12481	. . 3 |- ( ( J e. Top /\ S C_ X ) -> ( ( int ` J ) ` S ) = U. ( J i^i ~P S ) )
14	7, 8, 13	syl2anc 409	. 2 |- ( ( J e. Top /\ S C_ X /\ T C_ S ) -> ( ( int ` J ) ` S ) = U. ( J i^i ~P S ) )
15	6, 12, 14	3sstr4d 3173	1 |- ( ( J e. Top /\ S C_ X /\ T C_ S ) -> ( ( int ` J ) ` T ) C_ ( ( int ` J ) ` S ) )

Syntax hints:   -> wi 4   /\ w3a 963   = wceq 1335   e. wcel 2128   i^i cin 3101   C_ wss 3102   ~Pcpw 3543   U.cuni 3772   ` cfv 5169   Topctop 12366   intcnt 12464

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-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-un 4393

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-id 4253  df-xp 4591  df-rel 4592  df-cnv 4593  df-co 4594  df-dm 4595  df-rn 4596  df-res 4597  df-ima 4598  df-iota 5134  df-fun 5171  df-fn 5172  df-f 5173  df-f1 5174  df-fo 5175  df-f1o 5176  df-fv 5177  df-top 12367  df-ntr 12467

This theorem is referenced by:  ntrin  12495  ntrcls0  12502