Theorem ntrval 18782

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 18770	. . . 4 |- ( J e. Top -> ( int ` J ) = ( x e. ~P X |-> U. ( J i^i ~P x ) ) )
3	2	fveq1d 5804	. . 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 18661	. . . . 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 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 4546	. . . . 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 6490	. . . 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 3974	. . . . . 6 |- ( x = S -> ~P x = ~P S )
14	13	ineq2d 3663	. . . . 5 |- ( x = S -> ( J i^i ~P x ) = ( J i^i ~P S ) )
15	14	unieqd 4212	. . . 4 |- ( x = S -> U. ( J i^i ~P x ) = U. ( J i^i ~P S ) )
16		eqid 2454	. . . 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 5884	. . 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 2495	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 1370   e. wcel 1758   _Vcvv 3078   i^i cin 3438   C_ wss 3439   ~Pcpw 3971   U.cuni 4202   |-> cmpt 4461   ` cfv 5529   Topctop 18640   intcnt 18763

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-top 18645  df-ntr 18766

This theorem is referenced by:  ntropn  18795  clsval2  18796  ntrss2  18803  ssntr  18804  isopn3  18812  ntreq0  18823