Theorem ntrval 19403

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 19391	. . . 4 |- ( J e. Top -> ( int ` J ) = ( x e. ~P X |-> U. ( J i^i ~P x ) ) )
3	2	fveq1d 5854	. . 3 |- ( J e. Top -> ( ( int ` J ) ` S ) = ( ( x e. ~P X |-> U. ( J i^i ~P x ) ) ` S ) )
4	3	adantr 465	. 2 |- ( ( J e. Top /\ S C_ X ) -> ( ( int ` J ) ` S ) = ( ( x e. ~P X |-> U. ( J i^i ~P x ) ) ` S ) )
5	1	topopn 19282	. . . . 5 |- ( J e. Top -> X e. J )
6		elpw2g 4596	. . . . 5 |- ( X e. J -> ( S e. ~P X <-> S C_ X ) )
7	5, 6	syl 16	. . . 4 |- ( J e. Top -> ( S e. ~P X <-> S C_ X ) )
8	7	biimpar 485	. . 3 |- ( ( J e. Top /\ S C_ X ) -> S e. ~P X )
9		inex1g 4576	. . . . 5 |- ( J e. Top -> ( J i^i ~P S ) e. _V )
10	9	adantr 465	. . . 4 |- ( ( J e. Top /\ S C_ X ) -> ( J i^i ~P S ) e. _V )
11		uniexg 6578	. . . 4 |- ( ( J i^i ~P S ) e. _V -> U. ( J i^i ~P S ) e. _V )
12	10, 11	syl 16	. . 3 |- ( ( J e. Top /\ S C_ X ) -> U. ( J i^i ~P S ) e. _V )
13		pweq 3996	. . . . . 6 |- ( x = S -> ~P x = ~P S )
14	13	ineq2d 3682	. . . . 5 |- ( x = S -> ( J i^i ~P x ) = ( J i^i ~P S ) )
15	14	unieqd 4240	. . . 4 |- ( x = S -> U. ( J i^i ~P x ) = U. ( J i^i ~P S ) )
16		eqid 2441	. . . 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 5935	. . 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 661	. 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 2482	1 |- ( ( J e. Top /\ S C_ X ) -> ( ( int ` J ) ` S ) = U. ( J i^i ~P S ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1381   e. wcel 1802   _Vcvv 3093   i^i cin 3457   C_ wss 3458   ~Pcpw 3993   U.cuni 4230   |-> cmpt 4491   ` cfv 5574   Topctop 19261   intcnt 19384

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-reu 2798  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-id 4781  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-top 19266  df-ntr 19387

This theorem is referenced by:  ntropn  19416  clsval2  19417  ntrss2  19424  ssntr  19425  isopn3  19433  ntreq0  19444