Theorem iscld 14777

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 14773	. . . 4 |- ( J e. Top -> ( Clsd ` J ) = { x e. ~P X | ( X \ x ) e. J } )
3	2	eleq2d 2299	. . 3 |- ( J e. Top -> ( S e. ( Clsd ` J ) <-> S e. { x e. ~P X | ( X \ x ) e. J } ) )
4		difeq2 3316	. . . . 5 |- ( x = S -> ( X \ x ) = ( X \ S ) )
5	4	eleq1d 2298	. . . 4 |- ( x = S -> ( ( X \ x ) e. J <-> ( X \ S ) e. J ) )
6	5	elrab 2959	. . 3 |- ( S e. { x e. ~P X | ( X \ x ) e. J } <-> ( S e. ~P X /\ ( X \ S ) e. J ) )
7	3, 6	bitrdi 196	. 2 |- ( J e. Top -> ( S e. ( Clsd ` J ) <-> ( S e. ~P X /\ ( X \ S ) e. J ) ) )
8	1	topopn 14682	. . . 4 |- ( J e. Top -> X e. J )
9		elpw2g 4240	. . . 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 465	. 2 |- ( J e. Top -> ( ( S e. ~P X /\ ( X \ S ) e. J ) <-> ( S C_ X /\ ( X \ S ) e. J ) ) )
12	7, 11	bitrd 188	1 |- ( J e. Top -> ( S e. ( Clsd ` J ) <-> ( S C_ X /\ ( X \ S ) e. J ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   {crab 2512   \ cdif 3194   C_ wss 3197   ~Pcpw 3649   U.cuni 3888   ` cfv 5318   Topctop 14671   Clsdccld 14766

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-in1 617  ax-in2 618  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-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

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-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-iota 5278  df-fun 5320  df-fv 5326  df-top 14672  df-cld 14769

This theorem is referenced by:  iscld2  14778  cldss  14779  cldopn  14781  topcld  14783  discld  14810