Theorem iscld 12514

Description: The predicate "the class S is a closed set". (Contributed by NM, 2-Oct-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)

Hypothesis

Ref	Expression
iscld.1	|- X = U. J

Assertion

Ref	Expression
iscld	|- ( J e. Top -> ( S e. ( Clsd ` J ) <-> ( S C_ X /\ ( X \ S ) e. J ) ) )

Proof of Theorem iscld

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		iscld.1	. . . . 5 |- X = U. J
2	1	cldval 12510	. . . 4 |- ( J e. Top -> ( Clsd ` J ) = { x e. ~P X | ( X \ x ) e. J } )
3	2	eleq2d 2227	. . 3 |- ( J e. Top -> ( S e. ( Clsd ` J ) <-> S e. { x e. ~P X | ( X \ x ) e. J } ) )
4		difeq2 3219	. . . . 5 |- ( x = S -> ( X \ x ) = ( X \ S ) )
5	4	eleq1d 2226	. . . 4 |- ( x = S -> ( ( X \ x ) e. J <-> ( X \ S ) e. J ) )
6	5	elrab 2868	. . 3 |- ( S e. { x e. ~P X | ( X \ x ) e. J } <-> ( S e. ~P X /\ ( X \ S ) e. J ) )
7	3, 6	bitrdi 195	. 2 |- ( J e. Top -> ( S e. ( Clsd ` J ) <-> ( S e. ~P X /\ ( X \ S ) e. J ) ) )
8	1	topopn 12417	. . . 4 |- ( J e. Top -> X e. J )
9		elpw2g 4117	. . . 4 |- ( X e. J -> ( S e. ~P X <-> S C_ X ) )
10	8, 9	syl 14	. . 3 |- ( J e. Top -> ( S e. ~P X <-> S C_ X ) )
11	10	anbi1d 461	. 2 |- ( J e. Top -> ( ( S e. ~P X /\ ( X \ S ) e. J ) <-> ( S C_ X /\ ( X \ S ) e. J ) ) )
12	7, 11	bitrd 187	1 |- ( J e. Top -> ( S e. ( Clsd ` J ) <-> ( S C_ X /\ ( X \ S ) e. J ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1335   e. wcel 2128   {crab 2439   \ cdif 3099   C_ wss 3102   ~Pcpw 3543   U.cuni 3772   ` cfv 5170   Topctop 12406   Clsdccld 12503

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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-sbc 2938  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-mpt 4027  df-id 4253  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-iota 5135  df-fun 5172  df-fv 5178  df-top 12407  df-cld 12506

This theorem is referenced by:  iscld2  12515  cldss  12516  cldopn  12518  topcld  12520  discld  12547