Theorem flimval 21767

Description: The set of limit points of a filter. (Contributed by Jeff Hankins, 4-Sep-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)

Hypothesis

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

Assertion

Ref	Expression
flimval	|- ( ( J e. Top /\ F e. U. ran Fil ) -> ( J fLim F ) = { x e. X | ( ( ( nei ` J ) ` { x } ) C_ F /\ F C_ ~P X ) } )

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

Proof of Theorem flimval

Dummy variables f j are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		flimval.1	. . . . 5 |- X = U. J
2	1	topopn 20711	. . . 4 |- ( J e. Top -> X e. J )
3	2	adantr 481	. . 3 |- ( ( J e. Top /\ F e. U. ran Fil ) -> X e. J )
4		rabexg 4812	. . 3 |- ( X e. J -> { x e. X | ( ( ( nei ` J ) ` { x } ) C_ F /\ F C_ ~P X ) } e. _V )
5	3, 4	syl 17	. 2 |- ( ( J e. Top /\ F e. U. ran Fil ) -> { x e. X | ( ( ( nei ` J ) ` { x } ) C_ F /\ F C_ ~P X ) } e. _V )
6		simpl 473	. . . . . 6 |- ( ( j = J /\ f = F ) -> j = J )
7	6	unieqd 4446	. . . . 5 |- ( ( j = J /\ f = F ) -> U. j = U. J )
8	7, 1	syl6eqr 2674	. . . 4 |- ( ( j = J /\ f = F ) -> U. j = X )
9	6	fveq2d 6195	. . . . . . 7 |- ( ( j = J /\ f = F ) -> ( nei ` j ) = ( nei ` J ) )
10	9	fveq1d 6193	. . . . . 6 |- ( ( j = J /\ f = F ) -> ( ( nei ` j ) ` { x } ) = ( ( nei ` J ) ` { x } ) )
11		simpr 477	. . . . . 6 |- ( ( j = J /\ f = F ) -> f = F )
12	10, 11	sseq12d 3634	. . . . 5 |- ( ( j = J /\ f = F ) -> ( ( ( nei ` j ) ` { x } ) C_ f <-> ( ( nei ` J ) ` { x } ) C_ F ) )
13	8	pweqd 4163	. . . . . 6 |- ( ( j = J /\ f = F ) -> ~P U. j = ~P X )
14	11, 13	sseq12d 3634	. . . . 5 |- ( ( j = J /\ f = F ) -> ( f C_ ~P U. j <-> F C_ ~P X ) )
15	12, 14	anbi12d 747	. . . 4 |- ( ( j = J /\ f = F ) -> ( ( ( ( nei ` j ) ` { x } ) C_ f /\ f C_ ~P U. j ) <-> ( ( ( nei ` J ) ` { x } ) C_ F /\ F C_ ~P X ) ) )
16	8, 15	rabeqbidv 3195	. . 3 |- ( ( j = J /\ f = F ) -> { x e. U. j | ( ( ( nei ` j ) ` { x } ) C_ f /\ f C_ ~P U. j ) } = { x e. X | ( ( ( nei ` J ) ` { x } ) C_ F /\ F C_ ~P X ) } )
17		df-flim 21743	. . 3 |- fLim = ( j e. Top , f e. U. ran Fil |-> { x e. U. j | ( ( ( nei ` j ) ` { x } ) C_ f /\ f C_ ~P U. j ) } )
18	16, 17	ovmpt2ga 6790	. 2 |- ( ( J e. Top /\ F e. U. ran Fil /\ { x e. X | ( ( ( nei ` J ) ` { x } ) C_ F /\ F C_ ~P X ) } e. _V ) -> ( J fLim F ) = { x e. X | ( ( ( nei ` J ) ` { x } ) C_ F /\ F C_ ~P X ) } )
19	5, 18	mpd3an3 1425	1 |- ( ( J e. Top /\ F e. U. ran Fil ) -> ( J fLim F ) = { x e. X | ( ( ( nei ` J ) ` { x } ) C_ F /\ F C_ ~P X ) } )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916   _Vcvv 3200   C_ wss 3574   ~Pcpw 4158   {csn 4177   U.cuni 4436   ran crn 5115   ` cfv 5888  (class class class)co 6650   Topctop 20698   neicnei 20901   Filcfil 21649   fLim cflim 21738

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-sep 4781  ax-nul 4789  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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  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-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-top 20699  df-flim 21743

This theorem is referenced by:  elflim2  21768