Theorem ntrfval 14642

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 14550	. . 3 |- ( J e. Top -> X e. J )
3		pwexg 4231	. . 3 |- ( X e. J -> ~P X e. _V )
4		mptexg 5821	. . 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 3864	. . . . . 6 |- ( j = J -> U. j = U. J )
7	6, 1	eqtr4di 2257	. . . . 5 |- ( j = J -> U. j = X )
8	7	pweqd 3625	. . . 4 |- ( j = J -> ~P U. j = ~P X )
9		ineq1 3371	. . . . 5 |- ( j = J -> ( j i^i ~P x ) = ( J i^i ~P x ) )
10	9	unieqd 3866	. . . 4 |- ( j = J -> U. ( j i^i ~P x ) = U. ( J i^i ~P x ) )
11	8, 10	mpteq12dv 4133	. . 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 14638	. . 3 |- int = ( j e. Top |-> ( x e. ~P U. j |-> U. ( j i^i ~P x ) ) )
13	11, 12	fvmptg 5667	. 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 1373   e. wcel 2177   _Vcvv 2773   i^i cin 3169   ~Pcpw 3620   U.cuni 3855   |-> cmpt 4112   ` cfv 5279   Topctop 14539   intcnt 14635

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2180  ax-ext 2188  ax-coll 4166  ax-sep 4169  ax-pow 4225  ax-pr 4260

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-un 3174  df-in 3176  df-ss 3183  df-pw 3622  df-sn 3643  df-pr 3644  df-op 3646  df-uni 3856  df-iun 3934  df-br 4051  df-opab 4113  df-mpt 4114  df-id 4347  df-xp 4688  df-rel 4689  df-cnv 4690  df-co 4691  df-dm 4692  df-rn 4693  df-res 4694  df-ima 4695  df-iota 5240  df-fun 5281  df-fn 5282  df-f 5283  df-f1 5284  df-fo 5285  df-f1o 5286  df-fv 5287  df-top 14540  df-ntr 14638

This theorem is referenced by:  ntrval  14652