Theorem fcfneii 21841

Description: A neighborhood of a cluster point of a function contains a function value from every tail. (Contributed by Jeff Hankins, 27-Nov-2009.) (Revised by Stefan O'Rear, 9-Aug-2015.)

Assertion

Ref	Expression
fcfneii	|- ( ( ( J e. (TopOn ` X ) /\ L e. ( Fil ` Y ) /\ F : Y --> X ) /\ ( A e. ( ( J fClusf L ) ` F ) /\ N e. ( ( nei ` J ) ` { A } ) /\ S e. L ) ) -> ( N i^i ( F " S ) ) =/= (/) )

Proof of Theorem fcfneii

Dummy variables n s are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fcfnei 21839	. . 3 |- ( ( J e. (TopOn ` X ) /\ L e. ( Fil ` Y ) /\ F : Y --> X ) -> ( A e. ( ( J fClusf L ) ` F ) <-> ( A e. X /\ A. n e. ( ( nei ` J ) ` { A } ) A. s e. L ( n i^i ( F " s ) ) =/= (/) ) ) )
2		ineq1 3807	. . . . . . . 8 |- ( n = N -> ( n i^i ( F " s ) ) = ( N i^i ( F " s ) ) )
3	2	neeq1d 2853	. . . . . . 7 |- ( n = N -> ( ( n i^i ( F " s ) ) =/= (/) <-> ( N i^i ( F " s ) ) =/= (/) ) )
4		imaeq2 5462	. . . . . . . . 9 |- ( s = S -> ( F " s ) = ( F " S ) )
5	4	ineq2d 3814	. . . . . . . 8 |- ( s = S -> ( N i^i ( F " s ) ) = ( N i^i ( F " S ) ) )
6	5	neeq1d 2853	. . . . . . 7 |- ( s = S -> ( ( N i^i ( F " s ) ) =/= (/) <-> ( N i^i ( F " S ) ) =/= (/) ) )
7	3, 6	rspc2v 3322	. . . . . 6 |- ( ( N e. ( ( nei ` J ) ` { A } ) /\ S e. L ) -> ( A. n e. ( ( nei ` J ) ` { A } ) A. s e. L ( n i^i ( F " s ) ) =/= (/) -> ( N i^i ( F " S ) ) =/= (/) ) )
8	7	ex 450	. . . . 5 |- ( N e. ( ( nei ` J ) ` { A } ) -> ( S e. L -> ( A. n e. ( ( nei ` J ) ` { A } ) A. s e. L ( n i^i ( F " s ) ) =/= (/) -> ( N i^i ( F " S ) ) =/= (/) ) ) )
9	8	com3r 87	. . . 4 |- ( A. n e. ( ( nei ` J ) ` { A } ) A. s e. L ( n i^i ( F " s ) ) =/= (/) -> ( N e. ( ( nei ` J ) ` { A } ) -> ( S e. L -> ( N i^i ( F " S ) ) =/= (/) ) ) )
10	9	adantl 482	. . 3 |- ( ( A e. X /\ A. n e. ( ( nei ` J ) ` { A } ) A. s e. L ( n i^i ( F " s ) ) =/= (/) ) -> ( N e. ( ( nei ` J ) ` { A } ) -> ( S e. L -> ( N i^i ( F " S ) ) =/= (/) ) ) )
11	1, 10	syl6bi 243	. 2 |- ( ( J e. (TopOn ` X ) /\ L e. ( Fil ` Y ) /\ F : Y --> X ) -> ( A e. ( ( J fClusf L ) ` F ) -> ( N e. ( ( nei ` J ) ` { A } ) -> ( S e. L -> ( N i^i ( F " S ) ) =/= (/) ) ) ) )
12	11	3imp2 1282	1 |- ( ( ( J e. (TopOn ` X ) /\ L e. ( Fil ` Y ) /\ F : Y --> X ) /\ ( A e. ( ( J fClusf L ) ` F ) /\ N e. ( ( nei ` J ) ` { A } ) /\ S e. L ) ) -> ( N i^i ( F " S ) ) =/= (/) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   i^i cin 3573   (/)c0 3915   {csn 4177   "cima 5117   -->wf 5884   ` cfv 5888  (class class class)co 6650  TopOnctopon 20715   neicnei 20901   Filcfil 21649   fClusf cfcf 21741

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-nel 2898  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-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-fbas 19743  df-fg 19744  df-top 20699  df-topon 20716  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-fil 21650  df-fm 21742  df-fcls 21745  df-fcf 21746

This theorem is referenced by: (None)