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Theorem ntrval 16605
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 16593 . . . 4  |-  ( J  e.  Top  ->  ( int `  J )  =  ( x  e.  ~P X  |->  U. ( J  i^i  ~P x ) ) )
32fveq1d 5379 . . 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 16484 . . . . 5  |-  ( J  e.  Top  ->  X  e.  J )
6 elpw2g 4063 . . . . 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 4054 . . . . 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 4408 . . . 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 3533 . . . . . 6  |-  ( x  =  S  ->  ~P x  =  ~P S
)
1413ineq2d 3278 . . . . 5  |-  ( x  =  S  ->  ( J  i^i  ~P x )  =  ( J  i^i  ~P S ) )
1514unieqd 3738 . . . 4  |-  ( x  =  S  ->  U. ( J  i^i  ~P x )  =  U. ( J  i^i  ~P S ) )
16 eqid 2253 . . . 4  |-  ( x  e.  ~P X  |->  U. ( J  i^i  ~P x ) )  =  ( x  e.  ~P X  |->  U. ( J  i^i  ~P x ) )
1715, 16fvmptg 5452 . . 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 2285 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 2727    i^i cin 3077    C_ wss 3078   ~Pcpw 3530   U.cuni 3727    e. cmpt 3974   ` cfv 4592   Topctop 16463   intcnt 16586
This theorem is referenced by:  ntropn  16618  clsval2  16619  ntrss2  16626  ssntr  16627  isopn3  16635  ntreq0  16646
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 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403
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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-top 16468  df-ntr 16589
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