Theorem fclsval 21812

Description: The set of all cluster points of a filter. (Contributed by Jeff Hankins, 10-Nov-2009.) (Revised by Stefan O'Rear, 8-Aug-2015.)

Hypothesis

Ref	Expression
fclsval.x	|- X = U. J

Assertion

Ref	Expression
fclsval	|- ( ( J e. Top /\ F e. ( Fil ` Y ) ) -> ( J fClus F ) = if ( X = Y , |^|_ t e. F ( ( cls ` J ) ` t ) , (/) ) )

Distinct variable groups:   t, F   t, J

Allowed substitution hints:   X( t)   Y( t)

Proof of Theorem fclsval

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

Step	Hyp	Ref	Expression
1		simpl 473	. . 3 |- ( ( J e. Top /\ F e. ( Fil ` Y ) ) -> J e. Top )
2		fvssunirn 6217	. . . . 5 |- ( Fil ` Y ) C_ U. ran Fil
3	2	sseli 3599	. . . 4 |- ( F e. ( Fil ` Y ) -> F e. U. ran Fil )
4	3	adantl 482	. . 3 |- ( ( J e. Top /\ F e. ( Fil ` Y ) ) -> F e. U. ran Fil )
5		filn0 21666	. . . . . 6 |- ( F e. ( Fil ` Y ) -> F =/= (/) )
6	5	adantl 482	. . . . 5 |- ( ( J e. Top /\ F e. ( Fil ` Y ) ) -> F =/= (/) )
7		fvex 6201	. . . . . 6 |- ( ( cls ` J ) ` t ) e. _V
8	7	rgenw 2924	. . . . 5 |- A. t e. F ( ( cls ` J ) ` t ) e. _V
9		iinexg 4824	. . . . 5 |- ( ( F =/= (/) /\ A. t e. F ( ( cls ` J ) ` t ) e. _V ) -> |^|_ t e. F ( ( cls ` J ) ` t ) e. _V )
10	6, 8, 9	sylancl 694	. . . 4 |- ( ( J e. Top /\ F e. ( Fil ` Y ) ) -> |^|_ t e. F ( ( cls ` J ) ` t ) e. _V )
11		0ex 4790	. . . 4 |- (/) e. _V
12		ifcl 4130	. . . 4 |- ( ( |^|_ t e. F ( ( cls ` J ) ` t ) e. _V /\ (/) e. _V ) -> if ( X = U. F , |^|_ t e. F ( ( cls ` J ) ` t ) , (/) ) e. _V )
13	10, 11, 12	sylancl 694	. . 3 |- ( ( J e. Top /\ F e. ( Fil ` Y ) ) -> if ( X = U. F , |^|_ t e. F ( ( cls ` J ) ` t ) , (/) ) e. _V )
14		unieq 4444	. . . . . . 7 |- ( j = J -> U. j = U. J )
15		fclsval.x	. . . . . . 7 |- X = U. J
16	14, 15	syl6eqr 2674	. . . . . 6 |- ( j = J -> U. j = X )
17		unieq 4444	. . . . . 6 |- ( f = F -> U. f = U. F )
18	16, 17	eqeqan12d 2638	. . . . 5 |- ( ( j = J /\ f = F ) -> ( U. j = U. f <-> X = U. F ) )
19		iineq1 4535	. . . . . . 7 |- ( f = F -> |^|_ t e. f ( ( cls ` j ) ` t ) = |^|_ t e. F ( ( cls ` j ) ` t ) )
20	19	adantl 482	. . . . . 6 |- ( ( j = J /\ f = F ) -> |^|_ t e. f ( ( cls ` j ) ` t ) = |^|_ t e. F ( ( cls ` j ) ` t ) )
21		simpll 790	. . . . . . . . 9 |- ( ( ( j = J /\ f = F ) /\ t e. F ) -> j = J )
22	21	fveq2d 6195	. . . . . . . 8 |- ( ( ( j = J /\ f = F ) /\ t e. F ) -> ( cls ` j ) = ( cls ` J ) )
23	22	fveq1d 6193	. . . . . . 7 |- ( ( ( j = J /\ f = F ) /\ t e. F ) -> ( ( cls ` j ) ` t ) = ( ( cls ` J ) ` t ) )
24	23	iineq2dv 4543	. . . . . 6 |- ( ( j = J /\ f = F ) -> |^|_ t e. F ( ( cls ` j ) ` t ) = |^|_ t e. F ( ( cls ` J ) ` t ) )
25	20, 24	eqtrd 2656	. . . . 5 |- ( ( j = J /\ f = F ) -> |^|_ t e. f ( ( cls ` j ) ` t ) = |^|_ t e. F ( ( cls ` J ) ` t ) )
26	18, 25	ifbieq1d 4109	. . . 4 |- ( ( j = J /\ f = F ) -> if ( U. j = U. f , |^|_ t e. f ( ( cls ` j ) ` t ) , (/) ) = if ( X = U. F , |^|_ t e. F ( ( cls ` J ) ` t ) , (/) ) )
27		df-fcls 21745	. . . 4 |- fClus = ( j e. Top , f e. U. ran Fil |-> if ( U. j = U. f , |^|_ t e. f ( ( cls ` j ) ` t ) , (/) ) )
28	26, 27	ovmpt2ga 6790	. . 3 |- ( ( J e. Top /\ F e. U. ran Fil /\ if ( X = U. F , |^|_ t e. F ( ( cls ` J ) ` t ) , (/) ) e. _V ) -> ( J fClus F ) = if ( X = U. F , |^|_ t e. F ( ( cls ` J ) ` t ) , (/) ) )
29	1, 4, 13, 28	syl3anc 1326	. 2 |- ( ( J e. Top /\ F e. ( Fil ` Y ) ) -> ( J fClus F ) = if ( X = U. F , |^|_ t e. F ( ( cls ` J ) ` t ) , (/) ) )
30		filunibas 21685	. . . . 5 |- ( F e. ( Fil ` Y ) -> U. F = Y )
31	30	eqeq2d 2632	. . . 4 |- ( F e. ( Fil ` Y ) -> ( X = U. F <-> X = Y ) )
32	31	adantl 482	. . 3 |- ( ( J e. Top /\ F e. ( Fil ` Y ) ) -> ( X = U. F <-> X = Y ) )
33	32	ifbid 4108	. 2 |- ( ( J e. Top /\ F e. ( Fil ` Y ) ) -> if ( X = U. F , |^|_ t e. F ( ( cls ` J ) ` t ) , (/) ) = if ( X = Y , |^|_ t e. F ( ( cls ` J ) ` t ) , (/) ) )
34	29, 33	eqtrd 2656	1 |- ( ( J e. Top /\ F e. ( Fil ` Y ) ) -> ( J fClus F ) = if ( X = Y , |^|_ t e. F ( ( cls ` J ) ` t ) , (/) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   _Vcvv 3200   (/)c0 3915   ifcif 4086   U.cuni 4436   |^|_ciin 4521   ran crn 5115   ` cfv 5888  (class class class)co 6650   Topctop 20698   clsccl 20822   Filcfil 21649   fClus cfcls 21740

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-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-nel 2898  df-ral 2917  df-rex 2918  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-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-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-fbas 19743  df-fil 21650  df-fcls 21745

This theorem is referenced by:  isfcls  21813  fclscmpi  21833