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Theorem ntrfval 19393
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 19284 . . 3  |-  ( J  e.  Top  ->  X  e.  J )
3 pwexg 4637 . . 3  |-  ( X  e.  J  ->  ~P X  e.  _V )
4 mptexg 6141 . . 3  |-  ( ~P X  e.  _V  ->  ( x  e.  ~P X  |-> 
U. ( J  i^i  ~P x ) )  e. 
_V )
52, 3, 43syl 20 . 2  |-  ( J  e.  Top  ->  (
x  e.  ~P X  |-> 
U. ( J  i^i  ~P x ) )  e. 
_V )
6 unieq 4259 . . . . . 6  |-  ( j  =  J  ->  U. j  =  U. J )
76, 1syl6eqr 2526 . . . . 5  |-  ( j  =  J  ->  U. j  =  X )
87pweqd 4021 . . . 4  |-  ( j  =  J  ->  ~P U. j  =  ~P X
)
9 ineq1 3698 . . . . 5  |-  ( j  =  J  ->  (
j  i^i  ~P x
)  =  ( J  i^i  ~P x ) )
109unieqd 4261 . . . 4  |-  ( j  =  J  ->  U. (
j  i^i  ~P x
)  =  U. ( J  i^i  ~P x ) )
118, 10mpteq12dv 4531 . . 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 19389 . . 3  |-  int  =  ( j  e.  Top  |->  ( x  e.  ~P U. j  |->  U. ( j  i^i 
~P x ) ) )
1311, 12fvmptg 5955 . 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 668 1  |-  ( J  e.  Top  ->  ( int `  J )  =  ( x  e.  ~P X  |->  U. ( J  i^i  ~P x ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767   _Vcvv 3118    i^i cin 3480   ~Pcpw 4016   U.cuni 4251    |-> cmpt 4511   ` cfv 5594   Topctop 19263   intcnt 19386
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-top 19268  df-ntr 19389
This theorem is referenced by:  ntrval  19405
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