Theorem ntrfval 12258

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 12164	. . 3 |- ( J e. Top -> X e. J )
3		pwexg 4099	. . 3 |- ( X e. J -> ~P X e. _V )
4		mptexg 5638	. . 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 3740	. . . . . 6 |- ( j = J -> U. j = U. J )
7	6, 1	syl6eqr 2188	. . . . 5 |- ( j = J -> U. j = X )
8	7	pweqd 3510	. . . 4 |- ( j = J -> ~P U. j = ~P X )
9		ineq1 3265	. . . . 5 |- ( j = J -> ( j i^i ~P x ) = ( J i^i ~P x ) )
10	9	unieqd 3742	. . . 4 |- ( j = J -> U. ( j i^i ~P x ) = U. ( J i^i ~P x ) )
11	8, 10	mpteq12dv 4005	. . 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 12254	. . 3 |- int = ( j e. Top |-> ( x e. ~P U. j |-> U. ( j i^i ~P x ) ) )
13	11, 12	fvmptg 5490	. 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 417	1 |- ( J e. Top -> ( int ` J ) = ( x e. ~P X |-> U. ( J i^i ~P x ) ) )

Syntax hints:   -> wi 4   = wceq 1331   e. wcel 1480   _Vcvv 2681   i^i cin 3065   ~Pcpw 3505   U.cuni 3731   |-> cmpt 3984   ` cfv 5118   Topctop 12153   intcnt 12251

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 2119  ax-coll 4038  ax-sep 4041  ax-pow 4093  ax-pr 4126

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-top 12154  df-ntr 12254

This theorem is referenced by:  ntrval  12268