Theorem ntrfval 12308

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 12214	. . 3 |- ( J e. Top -> X e. J )
3		pwexg 4112	. . 3 |- ( X e. J -> ~P X e. _V )
4		mptexg 5653	. . 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 3753	. . . . . 6 |- ( j = J -> U. j = U. J )
7	6, 1	eqtr4di 2191	. . . . 5 |- ( j = J -> U. j = X )
8	7	pweqd 3520	. . . 4 |- ( j = J -> ~P U. j = ~P X )
9		ineq1 3275	. . . . 5 |- ( j = J -> ( j i^i ~P x ) = ( J i^i ~P x ) )
10	9	unieqd 3755	. . . 4 |- ( j = J -> U. ( j i^i ~P x ) = U. ( J i^i ~P x ) )
11	8, 10	mpteq12dv 4018	. . 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 12304	. . 3 |- int = ( j e. Top |-> ( x e. ~P U. j |-> U. ( j i^i ~P x ) ) )
13	11, 12	fvmptg 5505	. 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 418	1 |- ( J e. Top -> ( int ` J ) = ( x e. ~P X |-> U. ( J i^i ~P x ) ) )

Syntax hints:   -> wi 4   = wceq 1332   e. wcel 1481   _Vcvv 2689   i^i cin 3075   ~Pcpw 3515   U.cuni 3744   |-> cmpt 3997   ` cfv 5131   Topctop 12203   intcnt 12301

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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-top 12204  df-ntr 12304

This theorem is referenced by:  ntrval  12318