Theorem ntrss 15143

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 1030	. . 3 |- ( ( J e. Top /\ S C_ X /\ T C_ S ) -> T C_ S )
2		sspwb 4351	. . . . 5 |- ( T C_ S <-> ~P T C_ ~P S )
3		sslin 3457	. . . . 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 3954	. . 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 1028	. . 3 |- ( ( J e. Top /\ S C_ X /\ T C_ S ) -> J e. Top )
8		simp2 1029	. . . 4 |- ( ( J e. Top /\ S C_ X /\ T C_ S ) -> S C_ X )
9	1, 8	sstrd 3258	. . 3 |- ( ( J e. Top /\ S C_ X /\ T C_ S ) -> T C_ X )
10		clscld.1	. . . 4 |- X = U. J
11	10	ntrval 15134	. . 3 |- ( ( J e. Top /\ T C_ X ) -> ( ( int ` J ) ` T ) = U. ( J i^i ~P T ) )
12	7, 9, 11	syl2anc 415	. 2 |- ( ( J e. Top /\ S C_ X /\ T C_ S ) -> ( ( int ` J ) ` T ) = U. ( J i^i ~P T ) )
13	10	ntrval 15134	. . 3 |- ( ( J e. Top /\ S C_ X ) -> ( ( int ` J ) ` S ) = U. ( J i^i ~P S ) )
14	7, 8, 13	syl2anc 415	. 2 |- ( ( J e. Top /\ S C_ X /\ T C_ S ) -> ( ( int ` J ) ` S ) = U. ( J i^i ~P S ) )
15	6, 12, 14	3sstr4d 3293	1 |- ( ( J e. Top /\ S C_ X /\ T C_ S ) -> ( ( int ` J ) ` T ) C_ ( ( int ` J ) ` S ) )

Syntax hints:   -> wi 4   /\ w3a 1009   = wceq 1402   e. wcel 2209   i^i cin 3219   C_ wss 3220   ~Pcpw 3685   U.cuni 3930   ` cfv 5372   Topctop 15021   intcnt 15117

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-top 15022  df-ntr 15120

This theorem is referenced by:  ntrin  15148  ntrcls0  15155