Theorem ntrfval 13462

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.

Step	Hyp	Ref	Expression
1		cldval.1	. . . 4 |- X = U. J
2	1	topopn 13368	. . 3 |- ( J e. Top -> X e. J )
3		pwexg 4179	. . 3 |- ( X e. J -> ~P X e. _V )
4		mptexg 5739	. . 3 |- ( ~P X e. _V -> ( x e. ~P X |-> U. ( J i^i ~P x ) ) e. _V )
5	2, 3, 4	3syl 17	. 2 |- ( J e. Top -> ( x e. ~P X |-> U. ( J i^i ~P x ) ) e. _V )
6		unieq 3818	. . . . . 6 |- ( j = J -> U. j = U. J )
7	6, 1	eqtr4di 2228	. . . . 5 |- ( j = J -> U. j = X )
8	7	pweqd 3580	. . . 4 |- ( j = J -> ~P U. j = ~P X )
9		ineq1 3329	. . . . 5 |- ( j = J -> ( j i^i ~P x ) = ( J i^i ~P x ) )
10	9	unieqd 3820	. . . 4 |- ( j = J -> U. ( j i^i ~P x ) = U. ( J i^i ~P x ) )
11	8, 10	mpteq12dv 4084	. . 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 13458	. . 3 |- int = ( j e. Top |-> ( x e. ~P U. j |-> U. ( j i^i ~P x ) ) )
13	11, 12	fvmptg 5590	. 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 ) ) )
14	5, 13	mpdan 421	1 |- ( J e. Top -> ( int ` J ) = ( x e. ~P X |-> U. ( J i^i ~P x ) ) )

Syntax hints:   -> wi 4   = wceq 1353   e. wcel 2148   _Vcvv 2737   i^i cin 3128   ~Pcpw 3575   U.cuni 3809   |-> cmpt 4063   ` cfv 5214   Topctop 13357   intcnt 13455

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 4117  ax-sep 4120  ax-pow 4173  ax-pr 4208

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 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-iun 3888  df-br 4003  df-opab 4064  df-mpt 4065  df-id 4292  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-res 4637  df-ima 4638  df-iota 5176  df-fun 5216  df-fn 5217  df-f 5218  df-f1 5219  df-fo 5220  df-f1o 5221  df-fv 5222  df-top 13358  df-ntr 13458

This theorem is referenced by:  ntrval  13472