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Theorem ntrss 12125
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
StepHypRef Expression
1 simp3 964 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X  /\  T  C_  S )  ->  T  C_  S )
2 sspwb 4096 . . . . 5  |-  ( T 
C_  S  <->  ~P T  C_ 
~P S )
3 sslin 3266 . . . . 5  |-  ( ~P T  C_  ~P S  ->  ( J  i^i  ~P T )  C_  ( J  i^i  ~P S ) )
42, 3sylbi 120 . . . 4  |-  ( T 
C_  S  ->  ( J  i^i  ~P T ) 
C_  ( J  i^i  ~P S ) )
54unissd 3724 . . 3  |-  ( T 
C_  S  ->  U. ( J  i^i  ~P T ) 
C_  U. ( J  i^i  ~P S ) )
61, 5syl 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 962 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X  /\  T  C_  S )  ->  J  e.  Top )
8 simp2 963 . . . 4  |-  ( ( J  e.  Top  /\  S  C_  X  /\  T  C_  S )  ->  S  C_  X )
91, 8sstrd 3071 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X  /\  T  C_  S )  ->  T  C_  X )
10 clscld.1 . . . 4  |-  X  = 
U. J
1110ntrval 12116 . . 3  |-  ( ( J  e.  Top  /\  T  C_  X )  -> 
( ( int `  J
) `  T )  =  U. ( J  i^i  ~P T ) )
127, 9, 11syl2anc 406 . 2  |-  ( ( J  e.  Top  /\  S  C_  X  /\  T  C_  S )  ->  (
( int `  J
) `  T )  =  U. ( J  i^i  ~P T ) )
1310ntrval 12116 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( int `  J
) `  S )  =  U. ( J  i^i  ~P S ) )
147, 8, 13syl2anc 406 . 2  |-  ( ( J  e.  Top  /\  S  C_  X  /\  T  C_  S )  ->  (
( int `  J
) `  S )  =  U. ( J  i^i  ~P S ) )
156, 12, 143sstr4d 3106 1  |-  ( ( J  e.  Top  /\  S  C_  X  /\  T  C_  S )  ->  (
( int `  J
) `  T )  C_  ( ( int `  J
) `  S )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 943    = wceq 1312    e. wcel 1461    i^i cin 3034    C_ wss 3035   ~Pcpw 3474   U.cuni 3700   ` cfv 5079   Topctop 12001   intcnt 12099
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-13 1472  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-coll 4001  ax-sep 4004  ax-pow 4056  ax-pr 4089  ax-un 4313
This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ral 2393  df-rex 2394  df-reu 2395  df-rab 2397  df-v 2657  df-sbc 2877  df-csb 2970  df-un 3039  df-in 3041  df-ss 3048  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-iun 3779  df-br 3894  df-opab 3948  df-mpt 3949  df-id 4173  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-rn 4508  df-res 4509  df-ima 4510  df-iota 5044  df-fun 5081  df-fn 5082  df-f 5083  df-f1 5084  df-fo 5085  df-f1o 5086  df-fv 5087  df-top 12002  df-ntr 12102
This theorem is referenced by:  ntrin  12130  ntrcls0  12137
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