Theorem ntrval 14778

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 14768	. . . 4 |- ( J e. Top -> ( int ` J ) = ( x e. ~P X |-> U. ( J i^i ~P x ) ) )
3	2	fveq1d 5628	. . 3 |- ( J e. Top -> ( ( int ` J ) ` S ) = ( ( x e. ~P X |-> U. ( J i^i ~P x ) ) ` S ) )
4	3	adantr 276	. 2 |- ( ( J e. Top /\ S C_ X ) -> ( ( int ` J ) ` S ) = ( ( x e. ~P X |-> U. ( J i^i ~P x ) ) ` S ) )
5		eqid 2229	. . 3 |- ( x e. ~P X |-> U. ( J i^i ~P x ) ) = ( x e. ~P X |-> U. ( J i^i ~P x ) )
6		pweq 3652	. . . . 5 |- ( x = S -> ~P x = ~P S )
7	6	ineq2d 3405	. . . 4 |- ( x = S -> ( J i^i ~P x ) = ( J i^i ~P S ) )
8	7	unieqd 3898	. . 3 |- ( x = S -> U. ( J i^i ~P x ) = U. ( J i^i ~P S ) )
9	1	topopn 14676	. . . . 5 |- ( J e. Top -> X e. J )
10		elpw2g 4239	. . . . 5 |- ( X e. J -> ( S e. ~P X <-> S C_ X ) )
11	9, 10	syl 14	. . . 4 |- ( J e. Top -> ( S e. ~P X <-> S C_ X ) )
12	11	biimpar 297	. . 3 |- ( ( J e. Top /\ S C_ X ) -> S e. ~P X )
13		inex1g 4219	. . . . 5 |- ( J e. Top -> ( J i^i ~P S ) e. _V )
14	13	adantr 276	. . . 4 |- ( ( J e. Top /\ S C_ X ) -> ( J i^i ~P S ) e. _V )
15		uniexg 4529	. . . 4 |- ( ( J i^i ~P S ) e. _V -> U. ( J i^i ~P S ) e. _V )
16	14, 15	syl 14	. . 3 |- ( ( J e. Top /\ S C_ X ) -> U. ( J i^i ~P S ) e. _V )
17	5, 8, 12, 16	fvmptd3 5727	. 2 |- ( ( J e. Top /\ S C_ X ) -> ( ( x e. ~P X |-> U. ( J i^i ~P x ) ) ` S ) = U. ( J i^i ~P S ) )
18	4, 17	eqtrd 2262	1 |- ( ( J e. Top /\ S C_ X ) -> ( ( int ` J ) ` S ) = U. ( J i^i ~P S ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   _Vcvv 2799   i^i cin 3196   C_ wss 3197   ~Pcpw 3649   U.cuni 3887   |-> cmpt 4144   ` cfv 5317   Topctop 14665   intcnt 14761

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4523

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-top 14666  df-ntr 14764

This theorem is referenced by:  ntropn  14785  ntrss  14787  ntrss2  14789  ssntr  14790  isopn3  14793  ntreq0  14800