Theorem clsval 19705

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 19693	. . . 4 |- ( J e. Top -> ( cls ` J ) = ( y e. ~P X |-> |^| { x e. ( Clsd ` J ) | y C_ x } ) )
3	2	fveq1d 5850	. . 3 |- ( J e. Top -> ( ( cls ` J ) ` S ) = ( ( y e. ~P X |-> |^| { x e. ( Clsd ` J ) | y C_ x } ) ` S ) )
4	3	adantr 463	. 2 |- ( ( J e. Top /\ S C_ X ) -> ( ( cls ` J ) ` S ) = ( ( y e. ~P X |-> |^| { x e. ( Clsd ` J ) | y C_ x } ) ` S ) )
5	1	topopn 19582	. . . . 5 |- ( J e. Top -> X e. J )
6		elpw2g 4600	. . . . 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 483	. . 3 |- ( ( J e. Top /\ S C_ X ) -> S e. ~P X )
9	1	topcld 19703	. . . . 5 |- ( J e. Top -> X e. ( Clsd ` J ) )
10		sseq2 3511	. . . . . 6 |- ( x = X -> ( S C_ x <-> S C_ X ) )
11	10	rspcev 3207	. . . . 5 |- ( ( X e. ( Clsd ` J ) /\ S C_ X ) -> E. x e. ( Clsd ` J ) S C_ x )
12	9, 11	sylan 469	. . . 4 |- ( ( J e. Top /\ S C_ X ) -> E. x e. ( Clsd ` J ) S C_ x )
13		intexrab 4596	. . . 4 |- ( E. x e. ( Clsd ` J ) S C_ x <-> |^| { x e. ( Clsd ` J ) | S C_ x } e. _V )
14	12, 13	sylib 196	. . 3 |- ( ( J e. Top /\ S C_ X ) -> |^| { x e. ( Clsd ` J ) | S C_ x } e. _V )
15		sseq1 3510	. . . . . 6 |- ( y = S -> ( y C_ x <-> S C_ x ) )
16	15	rabbidv 3098	. . . . 5 |- ( y = S -> { x e. ( Clsd ` J ) | y C_ x } = { x e. ( Clsd ` J ) | S C_ x } )
17	16	inteqd 4276	. . . 4 |- ( y = S -> |^| { x e. ( Clsd ` J ) | y C_ x } = |^| { x e. ( Clsd ` J ) | S C_ x } )
18		eqid 2454	. . . 4 |- ( y e. ~P X |-> |^| { x e. ( Clsd ` J ) | y C_ x } ) = ( y e. ~P X |-> |^| { x e. ( Clsd ` J ) | y C_ x } )
19	17, 18	fvmptg 5929	. . 3 |- ( ( S e. ~P X /\ |^| { x e. ( Clsd ` J ) | S C_ x } e. _V ) -> ( ( y e. ~P X |-> |^| { x e. ( Clsd ` J ) | y C_ x } ) ` S ) = |^| { x e. ( Clsd ` J ) | S C_ x } )
20	8, 14, 19	syl2anc 659	. 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 } )
21	4, 20	eqtrd 2495	1 |- ( ( J e. Top /\ S C_ X ) -> ( ( cls ` J ) ` S ) = |^| { x e. ( Clsd ` J ) | S C_ x } )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 367   = wceq 1398   e. wcel 1823   E.wrex 2805   {crab 2808   _Vcvv 3106   C_ wss 3461   ~Pcpw 3999   U.cuni 4235   |^|cint 4271   |-> cmpt 4497   ` cfv 5570   Topctop 19561   Clsdccld 19684   clsccl 19686

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-top 19566  df-cld 19687  df-cls 19689

This theorem is referenced by:  cldcls  19710  clscld  19715  clsf  19716  clsval2  19718  clsss  19722  sscls  19724