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Theorem ntrval 16990
Description: The interior of a subset of a topology's base set is the union of all the open sets it includes. Definition of interior of [Munkres] p. 94. (Contributed by NM, 10-Sep-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)
Hypothesis
Ref Expression
iscld.1  |-  X  = 
U. J
Assertion
Ref Expression
ntrval  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( int `  J
) `  S )  =  U. ( J  i^i  ~P S ) )

Proof of Theorem ntrval
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 iscld.1 . . . . 5  |-  X  = 
U. J
21ntrfval 16978 . . . 4  |-  ( J  e.  Top  ->  ( int `  J )  =  ( x  e.  ~P X  |->  U. ( J  i^i  ~P x ) ) )
32fveq1d 5634 . . 3  |-  ( J  e.  Top  ->  (
( int `  J
) `  S )  =  ( ( x  e.  ~P X  |->  U. ( J  i^i  ~P x ) ) `  S ) )
43adantr 451 . 2  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( int `  J
) `  S )  =  ( ( x  e.  ~P X  |->  U. ( J  i^i  ~P x ) ) `  S ) )
51topopn 16869 . . . . 5  |-  ( J  e.  Top  ->  X  e.  J )
6 elpw2g 4276 . . . . 5  |-  ( X  e.  J  ->  ( S  e.  ~P X  <->  S 
C_  X ) )
75, 6syl 15 . . . 4  |-  ( J  e.  Top  ->  ( S  e.  ~P X  <->  S 
C_  X ) )
87biimpar 471 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X )  ->  S  e.  ~P X
)
9 inex1g 4259 . . . . 5  |-  ( J  e.  Top  ->  ( J  i^i  ~P S )  e.  _V )
109adantr 451 . . . 4  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( J  i^i  ~P S )  e.  _V )
11 uniexg 4620 . . . 4  |-  ( ( J  i^i  ~P S
)  e.  _V  ->  U. ( J  i^i  ~P S )  e.  _V )
1210, 11syl 15 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X )  ->  U. ( J  i^i  ~P S )  e.  _V )
13 pweq 3717 . . . . . 6  |-  ( x  =  S  ->  ~P x  =  ~P S
)
1413ineq2d 3458 . . . . 5  |-  ( x  =  S  ->  ( J  i^i  ~P x )  =  ( J  i^i  ~P S ) )
1514unieqd 3940 . . . 4  |-  ( x  =  S  ->  U. ( J  i^i  ~P x )  =  U. ( J  i^i  ~P S ) )
16 eqid 2366 . . . 4  |-  ( x  e.  ~P X  |->  U. ( J  i^i  ~P x ) )  =  ( x  e.  ~P X  |->  U. ( J  i^i  ~P x ) )
1715, 16fvmptg 5707 . . 3  |-  ( ( S  e.  ~P X  /\  U. ( J  i^i  ~P S )  e.  _V )  ->  ( ( x  e.  ~P X  |->  U. ( J  i^i  ~P x ) ) `  S )  =  U. ( J  i^i  ~P S
) )
188, 12, 17syl2anc 642 . 2  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( x  e. 
~P X  |->  U. ( J  i^i  ~P x ) ) `  S )  =  U. ( J  i^i  ~P S ) )
194, 18eqtrd 2398 1  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( int `  J
) `  S )  =  U. ( J  i^i  ~P S ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1647    e. wcel 1715   _Vcvv 2873    i^i cin 3237    C_ wss 3238   ~Pcpw 3714   U.cuni 3929    e. cmpt 4179   ` cfv 5358   Topctop 16848   intcnt 16971
This theorem is referenced by:  ntropn  17003  clsval2  17004  ntrss2  17011  ssntr  17012  isopn3  17020  ntreq0  17031
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-reu 2635  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-top 16853  df-ntr 16974
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