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Theorem ntrfval 17088
Description: The interior function on the subsets of a topology's base set. (Contributed by NM, 10-Sep-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)
Hypothesis
Ref Expression
cldval.1  |-  X  = 
U. J
Assertion
Ref Expression
ntrfval  |-  ( J  e.  Top  ->  ( int `  J )  =  ( x  e.  ~P X  |->  U. ( J  i^i  ~P x ) ) )
Distinct variable groups:    x, J    x, X

Proof of Theorem ntrfval
Dummy variable  j is distinct from all other variables.
StepHypRef Expression
1 cldval.1 . . . 4  |-  X  = 
U. J
21topopn 16979 . . 3  |-  ( J  e.  Top  ->  X  e.  J )
3 pwexg 4383 . . 3  |-  ( X  e.  J  ->  ~P X  e.  _V )
4 mptexg 5965 . . 3  |-  ( ~P X  e.  _V  ->  ( x  e.  ~P X  |-> 
U. ( J  i^i  ~P x ) )  e. 
_V )
52, 3, 43syl 19 . 2  |-  ( J  e.  Top  ->  (
x  e.  ~P X  |-> 
U. ( J  i^i  ~P x ) )  e. 
_V )
6 unieq 4024 . . . . . 6  |-  ( j  =  J  ->  U. j  =  U. J )
76, 1syl6eqr 2486 . . . . 5  |-  ( j  =  J  ->  U. j  =  X )
87pweqd 3804 . . . 4  |-  ( j  =  J  ->  ~P U. j  =  ~P X
)
9 ineq1 3535 . . . . 5  |-  ( j  =  J  ->  (
j  i^i  ~P x
)  =  ( J  i^i  ~P x ) )
109unieqd 4026 . . . 4  |-  ( j  =  J  ->  U. (
j  i^i  ~P x
)  =  U. ( J  i^i  ~P x ) )
118, 10mpteq12dv 4287 . . 3  |-  ( j  =  J  ->  (
x  e.  ~P U. j  |->  U. ( j  i^i 
~P x ) )  =  ( x  e. 
~P X  |->  U. ( J  i^i  ~P x ) ) )
12 df-ntr 17084 . . 3  |-  int  =  ( j  e.  Top  |->  ( x  e.  ~P U. j  |->  U. ( j  i^i 
~P x ) ) )
1311, 12fvmptg 5804 . 2  |-  ( ( J  e.  Top  /\  ( x  e.  ~P X  |->  U. ( J  i^i  ~P x ) )  e. 
_V )  ->  ( int `  J )  =  ( x  e.  ~P X  |->  U. ( J  i^i  ~P x ) ) )
145, 13mpdan 650 1  |-  ( J  e.  Top  ->  ( int `  J )  =  ( x  e.  ~P X  |->  U. ( J  i^i  ~P x ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   _Vcvv 2956    i^i cin 3319   ~Pcpw 3799   U.cuni 4015    e. cmpt 4266   ` cfv 5454   Topctop 16958   intcnt 17081
This theorem is referenced by:  ntrval  17100
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-top 16963  df-ntr 17084
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