Theorem clsval 12269

Description: The closure of a subset of a topology's base set is the intersection of all the closed sets that include it. Definition of closure 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
clsval	|- ( ( J e. Top /\ S C_ X ) -> ( ( cls ` J ) ` S ) = |^| { x e. ( Clsd ` J ) | S C_ x } )

Distinct variable groups:   x, J   x, S   x, X

Proof of Theorem clsval

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		iscld.1	. . . . 5 |- X = U. J
2	1	clsfval 12259	. . . 4 |- ( J e. Top -> ( cls ` J ) = ( y e. ~P X |-> |^| { x e. ( Clsd ` J ) | y C_ x } ) )
3	2	fveq1d 5416	. . 3 |- ( J e. Top -> ( ( cls ` J ) ` S ) = ( ( y e. ~P X |-> |^| { x e. ( Clsd ` J ) | y C_ x } ) ` S ) )
4	3	adantr 274	. 2 |- ( ( J e. Top /\ S C_ X ) -> ( ( cls ` J ) ` S ) = ( ( y e. ~P X |-> |^| { x e. ( Clsd ` J ) | y C_ x } ) ` S ) )
5		eqid 2137	. . 3 |- ( y e. ~P X |-> |^| { x e. ( Clsd ` J ) | y C_ x } ) = ( y e. ~P X |-> |^| { x e. ( Clsd ` J ) | y C_ x } )
6		sseq1 3115	. . . . 5 |- ( y = S -> ( y C_ x <-> S C_ x ) )
7	6	rabbidv 2670	. . . 4 |- ( y = S -> { x e. ( Clsd ` J ) | y C_ x } = { x e. ( Clsd ` J ) | S C_ x } )
8	7	inteqd 3771	. . 3 |- ( y = S -> |^| { x e. ( Clsd ` J ) | y C_ x } = |^| { x e. ( Clsd ` J ) | S C_ x } )
9	1	topopn 12164	. . . . 5 |- ( J e. Top -> X e. J )
10		elpw2g 4076	. . . . 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 295	. . 3 |- ( ( J e. Top /\ S C_ X ) -> S e. ~P X )
13	1	topcld 12267	. . . . 5 |- ( J e. Top -> X e. ( Clsd ` J ) )
14		sseq2 3116	. . . . . 6 |- ( x = X -> ( S C_ x <-> S C_ X ) )
15	14	rspcev 2784	. . . . 5 |- ( ( X e. ( Clsd ` J ) /\ S C_ X ) -> E. x e. ( Clsd ` J ) S C_ x )
16	13, 15	sylan 281	. . . 4 |- ( ( J e. Top /\ S C_ X ) -> E. x e. ( Clsd ` J ) S C_ x )
17		intexrabim 4073	. . . 4 |- ( E. x e. ( Clsd ` J ) S C_ x -> |^| { x e. ( Clsd ` J ) | S C_ x } e. _V )
18	16, 17	syl 14	. . 3 |- ( ( J e. Top /\ S C_ X ) -> |^| { x e. ( Clsd ` J ) | S C_ x } e. _V )
19	5, 8, 12, 18	fvmptd3 5507	. 2 |- ( ( J e. Top /\ S C_ X ) -> ( ( y e. ~P X |-> |^| { x e. ( Clsd ` J ) | y C_ x } ) ` S ) = |^| { x e. ( Clsd ` J ) | S C_ x } )
20	4, 19	eqtrd 2170	1 |- ( ( J e. Top /\ S C_ X ) -> ( ( cls ` J ) ` S ) = |^| { x e. ( Clsd ` J ) | S C_ x } )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1331   e. wcel 1480   E.wrex 2415   {crab 2418   _Vcvv 2681   C_ wss 3066   ~Pcpw 3505   U.cuni 3731   |^|cint 3766   |-> cmpt 3984   ` cfv 5118   Topctop 12153   Clsdccld 12250   clsccl 12252

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-pow 4093  ax-pr 4126

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-top 12154  df-cld 12253  df-cls 12255

This theorem is referenced by:  cldcls  12272  clsss  12276  sscls  12278