Theorem iscld 14339

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 14335	. . . 4 |- ( J e. Top -> ( Clsd ` J ) = { x e. ~P X | ( X \ x ) e. J } )
3	2	eleq2d 2266	. . 3 |- ( J e. Top -> ( S e. ( Clsd ` J ) <-> S e. { x e. ~P X | ( X \ x ) e. J } ) )
4		difeq2 3275	. . . . 5 |- ( x = S -> ( X \ x ) = ( X \ S ) )
5	4	eleq1d 2265	. . . 4 |- ( x = S -> ( ( X \ x ) e. J <-> ( X \ S ) e. J ) )
6	5	elrab 2920	. . 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 14244	. . . 4 |- ( J e. Top -> X e. J )
9		elpw2g 4189	. . . 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 1364   e. wcel 2167   {crab 2479   \ cdif 3154   C_ wss 3157   ~Pcpw 3605   U.cuni 3839   ` cfv 5258   Topctop 14233   Clsdccld 14328

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-iota 5219  df-fun 5260  df-fv 5266  df-top 14234  df-cld 14331

This theorem is referenced by:  iscld2  14340  cldss  14341  cldopn  14343  topcld  14345  discld  14372