Theorem lpfval 20942

Description: The limit point function on the subsets of a topology's base set. (Contributed by NM, 10-Feb-2007.) (Revised by Mario Carneiro, 11-Nov-2013.)

Hypothesis

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

Assertion

Ref	Expression
lpfval	|- ( J e. Top -> ( limPt ` J ) = ( x e. ~P X |-> { y | y e. ( ( cls ` J ) ` ( x \ { y } ) ) } ) )

Distinct variable groups:   x, y, J   x, X, y

Proof of Theorem lpfval

Dummy variable j is distinct from all other variables.

Step	Hyp	Ref	Expression
1		lpfval.1	. . . 4 |- X = U. J
2	1	topopn 20711	. . 3 |- ( J e. Top -> X e. J )
3		pwexg 4850	. . 3 |- ( X e. J -> ~P X e. _V )
4		mptexg 6484	. . 3 |- ( ~P X e. _V -> ( x e. ~P X |-> { y | y e. ( ( cls ` J ) ` ( x \ { y } ) ) } ) e. _V )
5	2, 3, 4	3syl 18	. 2 |- ( J e. Top -> ( x e. ~P X |-> { y | y e. ( ( cls ` J ) ` ( x \ { y } ) ) } ) e. _V )
6		unieq 4444	. . . . . 6 |- ( j = J -> U. j = U. J )
7	6, 1	syl6eqr 2674	. . . . 5 |- ( j = J -> U. j = X )
8	7	pweqd 4163	. . . 4 |- ( j = J -> ~P U. j = ~P X )
9		fveq2 6191	. . . . . . 7 |- ( j = J -> ( cls ` j ) = ( cls ` J ) )
10	9	fveq1d 6193	. . . . . 6 |- ( j = J -> ( ( cls ` j ) ` ( x \ { y } ) ) = ( ( cls ` J ) ` ( x \ { y } ) ) )
11	10	eleq2d 2687	. . . . 5 |- ( j = J -> ( y e. ( ( cls ` j ) ` ( x \ { y } ) ) <-> y e. ( ( cls ` J ) ` ( x \ { y } ) ) ) )
12	11	abbidv 2741	. . . 4 |- ( j = J -> { y | y e. ( ( cls ` j ) ` ( x \ { y } ) ) } = { y | y e. ( ( cls ` J ) ` ( x \ { y } ) ) } )
13	8, 12	mpteq12dv 4733	. . 3 |- ( j = J -> ( x e. ~P U. j |-> { y | y e. ( ( cls ` j ) ` ( x \ { y } ) ) } ) = ( x e. ~P X |-> { y | y e. ( ( cls ` J ) ` ( x \ { y } ) ) } ) )
14		df-lp 20940	. . 3 |- limPt = ( j e. Top |-> ( x e. ~P U. j |-> { y | y e. ( ( cls ` j ) ` ( x \ { y } ) ) } ) )
15	13, 14	fvmptg 6280	. 2 |- ( ( J e. Top /\ ( x e. ~P X |-> { y | y e. ( ( cls ` J ) ` ( x \ { y } ) ) } ) e. _V ) -> ( limPt ` J ) = ( x e. ~P X |-> { y | y e. ( ( cls ` J ) ` ( x \ { y } ) ) } ) )
16	5, 15	mpdan 702	1 |- ( J e. Top -> ( limPt ` J ) = ( x e. ~P X |-> { y | y e. ( ( cls ` J ) ` ( x \ { y } ) ) } ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   {cab 2608   _Vcvv 3200   \ cdif 3571   ~Pcpw 4158   {csn 4177   U.cuni 4436   |-> cmpt 4729   ` 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-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

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-iun 4522  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-lp 20940

This theorem is referenced by:  lpval  20943