Theorem ntrval 20128

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 20116	. . . 4 |- ( J e. Top -> ( int ` J ) = ( x e. ~P X |-> U. ( J i^i ~P x ) ) )
3	2	fveq1d 5881	. . 3 |- ( J e. Top -> ( ( int ` J ) ` S ) = ( ( x e. ~P X |-> U. ( J i^i ~P x ) ) ` S ) )
4	3	adantr 472	. 2 |- ( ( J e. Top /\ S C_ X ) -> ( ( int ` J ) ` S ) = ( ( x e. ~P X |-> U. ( J i^i ~P x ) ) ` S ) )
5	1	topopn 20013	. . . . 5 |- ( J e. Top -> X e. J )
6		elpw2g 4564	. . . . 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 493	. . 3 |- ( ( J e. Top /\ S C_ X ) -> S e. ~P X )
9		inex1g 4539	. . . . 5 |- ( J e. Top -> ( J i^i ~P S ) e. _V )
10	9	adantr 472	. . . 4 |- ( ( J e. Top /\ S C_ X ) -> ( J i^i ~P S ) e. _V )
11		uniexg 6607	. . . 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 3945	. . . . . 6 |- ( x = S -> ~P x = ~P S )
14	13	ineq2d 3625	. . . . 5 |- ( x = S -> ( J i^i ~P x ) = ( J i^i ~P S ) )
15	14	unieqd 4200	. . . 4 |- ( x = S -> U. ( J i^i ~P x ) = U. ( J i^i ~P S ) )
16		eqid 2471	. . . 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 5961	. . 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 673	. 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 2505	1 |- ( ( J e. Top /\ S C_ X ) -> ( ( int ` J ) ` S ) = U. ( J i^i ~P S ) )

Syntax hints:   -> wi 4   <-> wb 189   /\ wa 376   = wceq 1452   e. wcel 1904   _Vcvv 3031   i^i cin 3389   C_ wss 3390   ~Pcpw 3942   U.cuni 4190   |-> cmpt 4454   ` cfv 5589   Topctop 19994   intcnt 20109

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-top 19998  df-ntr 20112

This theorem is referenced by:  ntropn  20141  clsval2  20142  ntrss2  20149  ssntr  20150  isopn3  20159  ntreq0  20170