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Theorem ntrfval 13233
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 13139 . . 3  |-  ( J  e.  Top  ->  X  e.  J )
3 pwexg 4177 . . 3  |-  ( X  e.  J  ->  ~P X  e.  _V )
4 mptexg 5736 . . 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 3816 . . . . . 6  |-  ( j  =  J  ->  U. j  =  U. J )
76, 1eqtr4di 2228 . . . . 5  |-  ( j  =  J  ->  U. j  =  X )
87pweqd 3579 . . . 4  |-  ( j  =  J  ->  ~P U. j  =  ~P X
)
9 ineq1 3329 . . . . 5  |-  ( j  =  J  ->  (
j  i^i  ~P x
)  =  ( J  i^i  ~P x ) )
109unieqd 3818 . . . 4  |-  ( j  =  J  ->  U. (
j  i^i  ~P x
)  =  U. ( J  i^i  ~P x ) )
118, 10mpteq12dv 4082 . . 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 13229 . . 3  |-  int  =  ( j  e.  Top  |->  ( x  e.  ~P U. j  |->  U. ( j  i^i 
~P x ) ) )
1311, 12fvmptg 5587 . 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 421 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 1353    e. wcel 2148   _Vcvv 2737    i^i cin 3128   ~Pcpw 3574   U.cuni 3807    |-> cmpt 4061   ` cfv 5211   Topctop 13128   intcnt 13226
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-pow 4171  ax-pr 4205
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4289  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-ima 4635  df-iota 5173  df-fun 5213  df-fn 5214  df-f 5215  df-f1 5216  df-fo 5217  df-f1o 5218  df-fv 5219  df-top 13129  df-ntr 13229
This theorem is referenced by:  ntrval  13243
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