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Theorem ntrfval 12279
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 12185 . . 3  |-  ( J  e.  Top  ->  X  e.  J )
3 pwexg 4104 . . 3  |-  ( X  e.  J  ->  ~P X  e.  _V )
4 mptexg 5645 . . 3  |-  ( ~P X  e.  _V  ->  ( x  e.  ~P X  |-> 
U. ( J  i^i  ~P x ) )  e. 
_V )
52, 3, 43syl 17 . 2  |-  ( J  e.  Top  ->  (
x  e.  ~P X  |-> 
U. ( J  i^i  ~P x ) )  e. 
_V )
6 unieq 3745 . . . . . 6  |-  ( j  =  J  ->  U. j  =  U. J )
76, 1syl6eqr 2190 . . . . 5  |-  ( j  =  J  ->  U. j  =  X )
87pweqd 3515 . . . 4  |-  ( j  =  J  ->  ~P U. j  =  ~P X
)
9 ineq1 3270 . . . . 5  |-  ( j  =  J  ->  (
j  i^i  ~P x
)  =  ( J  i^i  ~P x ) )
109unieqd 3747 . . . 4  |-  ( j  =  J  ->  U. (
j  i^i  ~P x
)  =  U. ( J  i^i  ~P x ) )
118, 10mpteq12dv 4010 . . 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 12275 . . 3  |-  int  =  ( j  e.  Top  |->  ( x  e.  ~P U. j  |->  U. ( j  i^i 
~P x ) ) )
1311, 12fvmptg 5497 . 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 417 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 1331    e. wcel 1480   _Vcvv 2686    i^i cin 3070   ~Pcpw 3510   U.cuni 3736    |-> cmpt 3989   ` cfv 5123   Topctop 12174   intcnt 12272
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-pow 4098  ax-pr 4131
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-top 12175  df-ntr 12275
This theorem is referenced by:  ntrval  12289
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