Theorem ntrval 20051

Description: The interior of a subset of a topology's base set is the union of all the open sets it includes. Definition of interior of [Munkres] p. 94. (Contributed by NM, 10-Sep-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)

Hypothesis

Ref	Expression
iscld.1	|- X = U. J

Assertion

Ref	Expression
ntrval	|- ( ( J e. Top /\ S C_ X ) -> ( ( int ` J ) ` S ) = U. ( J i^i ~P S ) )

Proof of Theorem ntrval

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		iscld.1	. . . . 5 |- X = U. J
2	1	ntrfval 20039	. . . 4 |- ( J e. Top -> ( int ` J ) = ( x e. ~P X |-> U. ( J i^i ~P x ) ) )
3	2	fveq1d 5867	. . 3 |- ( J e. Top -> ( ( int ` J ) ` S ) = ( ( x e. ~P X |-> U. ( J i^i ~P x ) ) ` S ) )
4	3	adantr 467	. 2 |- ( ( J e. Top /\ S C_ X ) -> ( ( int ` J ) ` S ) = ( ( x e. ~P X |-> U. ( J i^i ~P x ) ) ` S ) )
5	1	topopn 19936	. . . . 5 |- ( J e. Top -> X e. J )
6		elpw2g 4566	. . . . 5 |- ( X e. J -> ( S e. ~P X <-> S C_ X ) )
7	5, 6	syl 17	. . . 4 |- ( J e. Top -> ( S e. ~P X <-> S C_ X ) )
8	7	biimpar 488	. . 3 |- ( ( J e. Top /\ S C_ X ) -> S e. ~P X )
9		inex1g 4546	. . . . 5 |- ( J e. Top -> ( J i^i ~P S ) e. _V )
10	9	adantr 467	. . . 4 |- ( ( J e. Top /\ S C_ X ) -> ( J i^i ~P S ) e. _V )
11		uniexg 6588	. . . 4 |- ( ( J i^i ~P S ) e. _V -> U. ( J i^i ~P S ) e. _V )
12	10, 11	syl 17	. . 3 |- ( ( J e. Top /\ S C_ X ) -> U. ( J i^i ~P S ) e. _V )
13		pweq 3954	. . . . . 6 |- ( x = S -> ~P x = ~P S )
14	13	ineq2d 3634	. . . . 5 |- ( x = S -> ( J i^i ~P x ) = ( J i^i ~P S ) )
15	14	unieqd 4208	. . . 4 |- ( x = S -> U. ( J i^i ~P x ) = U. ( J i^i ~P S ) )
16		eqid 2451	. . . 4 |- ( x e. ~P X |-> U. ( J i^i ~P x ) ) = ( x e. ~P X |-> U. ( J i^i ~P x ) )
17	15, 16	fvmptg 5946	. . 3 |- ( ( S e. ~P X /\ U. ( J i^i ~P S ) e. _V ) -> ( ( x e. ~P X |-> U. ( J i^i ~P x ) ) ` S ) = U. ( J i^i ~P S ) )
18	8, 12, 17	syl2anc 667	. 2 |- ( ( J e. Top /\ S C_ X ) -> ( ( x e. ~P X |-> U. ( J i^i ~P x ) ) ` S ) = U. ( J i^i ~P S ) )
19	4, 18	eqtrd 2485	1 |- ( ( J e. Top /\ S C_ X ) -> ( ( int ` J ) ` S ) = U. ( J i^i ~P S ) )

Syntax hints:   -> wi 4   <-> wb 188   /\ wa 371   = wceq 1444   e. wcel 1887   _Vcvv 3045   i^i cin 3403   C_ wss 3404   ~Pcpw 3951   U.cuni 4198   |-> cmpt 4461   ` cfv 5582   Topctop 19917   intcnt 20032

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-top 19921  df-ntr 20035

This theorem is referenced by:  ntropn  20064  clsval2  20065  ntrss2  20072  ssntr  20073  isopn3  20082  ntreq0  20093