Theorem ntrss 14641

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 1002	. . 3 |- ( ( J e. Top /\ S C_ X /\ T C_ S ) -> T C_ S )
2		sspwb 4265	. . . . 5 |- ( T C_ S <-> ~P T C_ ~P S )
3		sslin 3401	. . . . 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 3877	. . 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 1000	. . 3 |- ( ( J e. Top /\ S C_ X /\ T C_ S ) -> J e. Top )
8		simp2 1001	. . . 4 |- ( ( J e. Top /\ S C_ X /\ T C_ S ) -> S C_ X )
9	1, 8	sstrd 3205	. . 3 |- ( ( J e. Top /\ S C_ X /\ T C_ S ) -> T C_ X )
10		clscld.1	. . . 4 |- X = U. J
11	10	ntrval 14632	. . 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 14632	. . 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 3240	1 |- ( ( J e. Top /\ S C_ X /\ T C_ S ) -> ( ( int ` J ) ` T ) C_ ( ( int ` J ) ` S ) )

Syntax hints:   -> wi 4   /\ w3a 981   = wceq 1373   e. wcel 2177   i^i cin 3167   C_ wss 3168   ~Pcpw 3618   U.cuni 3853   ` cfv 5277   Topctop 14519   intcnt 14615

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4164  ax-sep 4167  ax-pow 4223  ax-pr 4258  ax-un 4485

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3001  df-csb 3096  df-un 3172  df-in 3174  df-ss 3181  df-pw 3620  df-sn 3641  df-pr 3642  df-op 3644  df-uni 3854  df-iun 3932  df-br 4049  df-opab 4111  df-mpt 4112  df-id 4345  df-xp 4686  df-rel 4687  df-cnv 4688  df-co 4689  df-dm 4690  df-rn 4691  df-res 4692  df-ima 4693  df-iota 5238  df-fun 5279  df-fn 5280  df-f 5281  df-f1 5282  df-fo 5283  df-f1o 5284  df-fv 5285  df-top 14520  df-ntr 14618

This theorem is referenced by:  ntrin  14646  ntrcls0  14653