Theorem ntrss 14778

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 1023	. . 3 |- ( ( J e. Top /\ S C_ X /\ T C_ S ) -> T C_ S )
2		sspwb 4301	. . . . 5 |- ( T C_ S <-> ~P T C_ ~P S )
3		sslin 3430	. . . . 5 |- ( ~P T C_ ~P S -> ( J i^i ~P T ) C_ ( J i^i ~P S ) )
4	2, 3	sylbi 121	. . . 4 |- ( T C_ S -> ( J i^i ~P T ) C_ ( J i^i ~P S ) )
5	4	unissd 3911	. . 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 1021	. . 3 |- ( ( J e. Top /\ S C_ X /\ T C_ S ) -> J e. Top )
8		simp2 1022	. . . 4 |- ( ( J e. Top /\ S C_ X /\ T C_ S ) -> S C_ X )
9	1, 8	sstrd 3234	. . 3 |- ( ( J e. Top /\ S C_ X /\ T C_ S ) -> T C_ X )
10		clscld.1	. . . 4 |- X = U. J
11	10	ntrval 14769	. . 3 |- ( ( J e. Top /\ T C_ X ) -> ( ( int ` J ) ` T ) = U. ( J i^i ~P T ) )
12	7, 9, 11	syl2anc 411	. 2 |- ( ( J e. Top /\ S C_ X /\ T C_ S ) -> ( ( int ` J ) ` T ) = U. ( J i^i ~P T ) )
13	10	ntrval 14769	. . 3 |- ( ( J e. Top /\ S C_ X ) -> ( ( int ` J ) ` S ) = U. ( J i^i ~P S ) )
14	7, 8, 13	syl2anc 411	. 2 |- ( ( J e. Top /\ S C_ X /\ T C_ S ) -> ( ( int ` J ) ` S ) = U. ( J i^i ~P S ) )
15	6, 12, 14	3sstr4d 3269	1 |- ( ( J e. Top /\ S C_ X /\ T C_ S ) -> ( ( int ` J ) ` T ) C_ ( ( int ` J ) ` S ) )

Syntax hints:   -> wi 4   /\ w3a 1002   = wceq 1395   e. wcel 2200   i^i cin 3196   C_ wss 3197   ~Pcpw 3649   U.cuni 3887   ` cfv 5314   Topctop 14656   intcnt 14752

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4521

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4381  df-xp 4722  df-rel 4723  df-cnv 4724  df-co 4725  df-dm 4726  df-rn 4727  df-res 4728  df-ima 4729  df-iota 5274  df-fun 5316  df-fn 5317  df-f 5318  df-f1 5319  df-fo 5320  df-f1o 5321  df-fv 5322  df-top 14657  df-ntr 14755

This theorem is referenced by:  ntrin  14783  ntrcls0  14790