Theorem islp 20944

Description: The predicate " P is a limit point of S." (Contributed by NM, 10-Feb-2007.)

Hypothesis

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

Assertion

Ref	Expression
islp	|- ( ( J e. Top /\ S C_ X ) -> ( P e. ( ( limPt ` J ) ` S ) <-> P e. ( ( cls ` J ) ` ( S \ { P } ) ) ) )

Proof of Theorem islp

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		lpfval.1	. . . 4 |- X = U. J
2	1	lpval 20943	. . 3 |- ( ( J e. Top /\ S C_ X ) -> ( ( limPt ` J ) ` S ) = { x | x e. ( ( cls ` J ) ` ( S \ { x } ) ) } )
3	2	eleq2d 2687	. 2 |- ( ( J e. Top /\ S C_ X ) -> ( P e. ( ( limPt ` J ) ` S ) <-> P e. { x | x e. ( ( cls ` J ) ` ( S \ { x } ) ) } ) )
4		elex 3212	. . 3 |- ( P e. ( ( cls ` J ) ` ( S \ { P } ) ) -> P e. _V )
5		id 22	. . . 4 |- ( x = P -> x = P )
6		sneq 4187	. . . . . 6 |- ( x = P -> { x } = { P } )
7	6	difeq2d 3728	. . . . 5 |- ( x = P -> ( S \ { x } ) = ( S \ { P } ) )
8	7	fveq2d 6195	. . . 4 |- ( x = P -> ( ( cls ` J ) ` ( S \ { x } ) ) = ( ( cls ` J ) ` ( S \ { P } ) ) )
9	5, 8	eleq12d 2695	. . 3 |- ( x = P -> ( x e. ( ( cls ` J ) ` ( S \ { x } ) ) <-> P e. ( ( cls ` J ) ` ( S \ { P } ) ) ) )
10	4, 9	elab3 3358	. 2 |- ( P e. { x | x e. ( ( cls ` J ) ` ( S \ { x } ) ) } <-> P e. ( ( cls ` J ) ` ( S \ { P } ) ) )
11	3, 10	syl6bb 276	1 |- ( ( J e. Top /\ S C_ X ) -> ( P e. ( ( limPt ` J ) ` S ) <-> P e. ( ( cls ` J ) ` ( S \ { P } ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   \ cdif 3571   C_ wss 3574   {csn 4177   U.cuni 4436   ` cfv 5888   Topctop 20698   clsccl 20822   limPtclp 20938

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-top 20699  df-cld 20823  df-cls 20825  df-lp 20940

This theorem is referenced by:  lpdifsn  20947  lpss3  20948  islp2  20949  islp3  20950  maxlp  20951  restlp  20987  lpcls  21168  limcnlp  23642  limcflflem  23644