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Theorem ntrval 16700
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
StepHypRef Expression
1 iscld.1 . . . . 5  |-  X  = 
U. J
21ntrfval 16688 . . . 4  |-  ( J  e.  Top  ->  ( int `  J )  =  ( x  e.  ~P X  |->  U. ( J  i^i  ~P x ) ) )
32fveq1d 5425 . . 3  |-  ( J  e.  Top  ->  (
( int `  J
) `  S )  =  ( ( x  e.  ~P X  |->  U. ( J  i^i  ~P x ) ) `  S ) )
43adantr 453 . 2  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( int `  J
) `  S )  =  ( ( x  e.  ~P X  |->  U. ( J  i^i  ~P x ) ) `  S ) )
51topopn 16579 . . . . 5  |-  ( J  e.  Top  ->  X  e.  J )
6 elpw2g 4107 . . . . 5  |-  ( X  e.  J  ->  ( S  e.  ~P X  <->  S 
C_  X ) )
75, 6syl 17 . . . 4  |-  ( J  e.  Top  ->  ( S  e.  ~P X  <->  S 
C_  X ) )
87biimpar 473 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X )  ->  S  e.  ~P X
)
9 inex1g 4097 . . . . 5  |-  ( J  e.  Top  ->  ( J  i^i  ~P S )  e.  _V )
109adantr 453 . . . 4  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( J  i^i  ~P S )  e.  _V )
11 uniexg 4454 . . . 4  |-  ( ( J  i^i  ~P S
)  e.  _V  ->  U. ( J  i^i  ~P S )  e.  _V )
1210, 11syl 17 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X )  ->  U. ( J  i^i  ~P S )  e.  _V )
13 pweq 3569 . . . . . 6  |-  ( x  =  S  ->  ~P x  =  ~P S
)
1413ineq2d 3312 . . . . 5  |-  ( x  =  S  ->  ( J  i^i  ~P x )  =  ( J  i^i  ~P S ) )
1514unieqd 3779 . . . 4  |-  ( x  =  S  ->  U. ( J  i^i  ~P x )  =  U. ( J  i^i  ~P S ) )
16 eqid 2256 . . . 4  |-  ( x  e.  ~P X  |->  U. ( J  i^i  ~P x ) )  =  ( x  e.  ~P X  |->  U. ( J  i^i  ~P x ) )
1715, 16fvmptg 5499 . . 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 645 . 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 2288 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 6    <-> wb 178    /\ wa 360    = wceq 1619    e. wcel 1621   _Vcvv 2740    i^i cin 3093    C_ wss 3094   ~Pcpw 3566   U.cuni 3768    e. cmpt 4017   ` cfv 4638   Topctop 16558   intcnt 16681
This theorem is referenced by:  ntropn  16713  clsval2  16714  ntrss2  16721  ssntr  16722  isopn3  16730  ntreq0  16741
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4071  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3769  df-iun 3848  df-br 3964  df-opab 4018  df-mpt 4019  df-id 4246  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-top 16563  df-ntr 16684
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