Theorem islp 7741

Description: The predicate "P is a limit point of S."

Hypothesis

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

Assertion

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

Proof of Theorem islp

Step	Hyp	Ref	Expression
1		lpfval.1	. . . 4 |- X = U.J
2	1	lpval 7740	. . 3 |- ((J e. Top /\ S (_ X) -> ((limPt` J)` S) = {x | x e. ((cls` J)` (S \ {x}))})
3	2	eleq2d 1544	. 2 |- ((J e. Top /\ S (_ X) -> (P e. ((limPt` J)` S) <-> P e. {x | x e. ((cls` J)` (S \ {x}))}))
4		elisset 1820	. . 3 |- (P e. ((cls` J)` (S \ {P})) -> P e. V)
5		id 59	. . . 4 |- (x = P -> x = P)
6		sneq 2421	. . . . . 6 |- (x = P -> {x} = {P})
7	6	difeq2d 2162	. . . . 5 |- (x = P -> (S \ {x}) = (S \ {P}))
8	7	fveq2d 3734	. . . 4 |- (x = P -> ((cls` J)` (S \ {x})) = ((cls` J)` (S \ {P})))
9	5, 8	eleq12d 1545	. . 3 |- (x = P -> (x e. ((cls` J)` (S \ {x})) <-> P e. ((cls` J)` (S \ {P}))))
10	4, 9	elab3 1906	. 2 |- (P e. {x | x e. ((cls` J)` (S \ {x}))} <-> P e. ((cls` J)` (S \ {P})))
11	3, 10	syl6bb 538	1 |- ((J e. Top /\ S (_ X) -> (P e. ((limPt` J)` S) <-> P e. ((cls` J)` (S \ {P}))))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 958   e. wcel 960  {cab 1466   \ cdif 2047   (_ wss 2050  {csn 2413  U.cuni 2507  ` cfv 3188  Topctop 7590  clsccl 7659  limPtclp 7737

This theorem is referenced by:  islp2 7744

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-rab 1655  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-int 2538  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-fv 3204  df-top 7594  df-cld 7660  df-cls 7662  df-lp 7738

Copyright terms: Public domain