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Theorem ntrval 18782
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 18770 . . . 4  |-  ( J  e.  Top  ->  ( int `  J )  =  ( x  e.  ~P X  |->  U. ( J  i^i  ~P x ) ) )
32fveq1d 5804 . . 3  |-  ( J  e.  Top  ->  (
( int `  J
) `  S )  =  ( ( x  e.  ~P X  |->  U. ( J  i^i  ~P x ) ) `  S ) )
43adantr 465 . 2  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( int `  J
) `  S )  =  ( ( x  e.  ~P X  |->  U. ( J  i^i  ~P x ) ) `  S ) )
51topopn 18661 . . . . 5  |-  ( J  e.  Top  ->  X  e.  J )
6 elpw2g 4566 . . . . 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 485 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X )  ->  S  e.  ~P X
)
9 inex1g 4546 . . . . 5  |-  ( J  e.  Top  ->  ( J  i^i  ~P S )  e.  _V )
109adantr 465 . . . 4  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( J  i^i  ~P S )  e.  _V )
11 uniexg 6490 . . . 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 3974 . . . . . 6  |-  ( x  =  S  ->  ~P x  =  ~P S
)
1413ineq2d 3663 . . . . 5  |-  ( x  =  S  ->  ( J  i^i  ~P x )  =  ( J  i^i  ~P S ) )
1514unieqd 4212 . . . 4  |-  ( x  =  S  ->  U. ( J  i^i  ~P x )  =  U. ( J  i^i  ~P S ) )
16 eqid 2454 . . . 4  |-  ( x  e.  ~P X  |->  U. ( J  i^i  ~P x ) )  =  ( x  e.  ~P X  |->  U. ( J  i^i  ~P x ) )
1715, 16fvmptg 5884 . . 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 661 . 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 2495 1  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( int `  J
) `  S )  =  U. ( J  i^i  ~P S ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758   _Vcvv 3078    i^i cin 3438    C_ wss 3439   ~Pcpw 3971   U.cuni 4202    |-> cmpt 4461   ` cfv 5529   Topctop 18640   intcnt 18763
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-top 18645  df-ntr 18766
This theorem is referenced by:  ntropn  18795  clsval2  18796  ntrss2  18803  ssntr  18804  isopn3  18812  ntreq0  18823
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