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Theorem ntrval 17055
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 17043 . . . 4  |-  ( J  e.  Top  ->  ( int `  J )  =  ( x  e.  ~P X  |->  U. ( J  i^i  ~P x ) ) )
32fveq1d 5689 . . 3  |-  ( J  e.  Top  ->  (
( int `  J
) `  S )  =  ( ( x  e.  ~P X  |->  U. ( J  i^i  ~P x ) ) `  S ) )
43adantr 452 . 2  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( int `  J
) `  S )  =  ( ( x  e.  ~P X  |->  U. ( J  i^i  ~P x ) ) `  S ) )
51topopn 16934 . . . . 5  |-  ( J  e.  Top  ->  X  e.  J )
6 elpw2g 4323 . . . . 5  |-  ( X  e.  J  ->  ( S  e.  ~P X  <->  S 
C_  X ) )
75, 6syl 16 . . . 4  |-  ( J  e.  Top  ->  ( S  e.  ~P X  <->  S 
C_  X ) )
87biimpar 472 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X )  ->  S  e.  ~P X
)
9 inex1g 4306 . . . . 5  |-  ( J  e.  Top  ->  ( J  i^i  ~P S )  e.  _V )
109adantr 452 . . . 4  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( J  i^i  ~P S )  e.  _V )
11 uniexg 4665 . . . 4  |-  ( ( J  i^i  ~P S
)  e.  _V  ->  U. ( J  i^i  ~P S )  e.  _V )
1210, 11syl 16 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X )  ->  U. ( J  i^i  ~P S )  e.  _V )
13 pweq 3762 . . . . . 6  |-  ( x  =  S  ->  ~P x  =  ~P S
)
1413ineq2d 3502 . . . . 5  |-  ( x  =  S  ->  ( J  i^i  ~P x )  =  ( J  i^i  ~P S ) )
1514unieqd 3986 . . . 4  |-  ( x  =  S  ->  U. ( J  i^i  ~P x )  =  U. ( J  i^i  ~P S ) )
16 eqid 2404 . . . 4  |-  ( x  e.  ~P X  |->  U. ( J  i^i  ~P x ) )  =  ( x  e.  ~P X  |->  U. ( J  i^i  ~P x ) )
1715, 16fvmptg 5763 . . 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 643 . 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 2436 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 177    /\ wa 359    = wceq 1649    e. wcel 1721   _Vcvv 2916    i^i cin 3279    C_ wss 3280   ~Pcpw 3759   U.cuni 3975    e. cmpt 4226   ` cfv 5413   Topctop 16913   intcnt 17036
This theorem is referenced by:  ntropn  17068  clsval2  17069  ntrss2  17076  ssntr  17077  isopn3  17085  ntreq0  17096
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-top 16918  df-ntr 17039
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