Theorem iscld 14575

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 14571	. . . 4 |- ( J e. Top -> ( Clsd ` J ) = { x e. ~P X | ( X \ x ) e. J } )
3	2	eleq2d 2275	. . 3 |- ( J e. Top -> ( S e. ( Clsd ` J ) <-> S e. { x e. ~P X | ( X \ x ) e. J } ) )
4		difeq2 3285	. . . . 5 |- ( x = S -> ( X \ x ) = ( X \ S ) )
5	4	eleq1d 2274	. . . 4 |- ( x = S -> ( ( X \ x ) e. J <-> ( X \ S ) e. J ) )
6	5	elrab 2929	. . 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 14480	. . . 4 |- ( J e. Top -> X e. J )
9		elpw2g 4200	. . . 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 1373   e. wcel 2176   {crab 2488   \ cdif 3163   C_ wss 3166   ~Pcpw 3616   U.cuni 3850   ` cfv 5271   Topctop 14469   Clsdccld 14564

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 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-rab 2493  df-v 2774  df-sbc 2999  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-iota 5232  df-fun 5273  df-fv 5279  df-top 14470  df-cld 14567

This theorem is referenced by:  iscld2  14576  cldss  14577  cldopn  14579  topcld  14581  discld  14608