Theorem lmcls 21106

Description: Any convergent sequence of points in a subset of a topological space converges to a point in the closure of the subset. (Contributed by Mario Carneiro, 30-Dec-2013.) (Revised by Mario Carneiro, 1-May-2014.)

Hypotheses

Ref	Expression
lmff.1	|- Z = ( ZZ>= ` M )
lmff.3	|- ( ph -> J e. (TopOn ` X ) )
lmff.4	|- ( ph -> M e. ZZ )
lmcls.5	|- ( ph -> F ( ~~> t ` J ) P )
lmcls.7	|- ( ( ph /\ k e. Z ) -> ( F ` k ) e. S )
lmcls.8	|- ( ph -> S C_ X )

Assertion

Ref	Expression
lmcls	|- ( ph -> P e. ( ( cls ` J ) ` S ) )

Distinct variable groups:   k, F   k, J   k, M   P, k   S, k   ph, k   k, X   k, Z

Proof of Theorem lmcls

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

Step	Hyp	Ref	Expression
1		lmcls.5	. . . . 5 |- ( ph -> F ( ~~> t ` J ) P )
2		lmff.3	. . . . . 6 |- ( ph -> J e. (TopOn ` X ) )
3		lmff.1	. . . . . 6 |- Z = ( ZZ>= ` M )
4		lmff.4	. . . . . 6 |- ( ph -> M e. ZZ )
5	2, 3, 4	lmbr2 21063	. . . . 5 |- ( ph -> ( F ( ~~> t ` J ) P <-> ( F e. ( X ^pm CC ) /\ P e. X /\ A. u e. J ( P e. u -> E. j e. Z A. k e. ( ZZ>= ` j ) ( k e. dom F /\ ( F ` k ) e. u ) ) ) ) )
6	1, 5	mpbid 222	. . . 4 |- ( ph -> ( F e. ( X ^pm CC ) /\ P e. X /\ A. u e. J ( P e. u -> E. j e. Z A. k e. ( ZZ>= ` j ) ( k e. dom F /\ ( F ` k ) e. u ) ) ) )
7	6	simp3d 1075	. . 3 |- ( ph -> A. u e. J ( P e. u -> E. j e. Z A. k e. ( ZZ>= ` j ) ( k e. dom F /\ ( F ` k ) e. u ) ) )
8	3	r19.2uz 14091	. . . . . 6 |- ( E. j e. Z A. k e. ( ZZ>= ` j ) ( k e. dom F /\ ( F ` k ) e. u ) -> E. k e. Z ( k e. dom F /\ ( F ` k ) e. u ) )
9		lmcls.7	. . . . . . . . 9 |- ( ( ph /\ k e. Z ) -> ( F ` k ) e. S )
10		inelcm 4032	. . . . . . . . . 10 |- ( ( ( F ` k ) e. u /\ ( F ` k ) e. S ) -> ( u i^i S ) =/= (/) )
11	10	a1i 11	. . . . . . . . 9 |- ( ( ph /\ k e. Z ) -> ( ( ( F ` k ) e. u /\ ( F ` k ) e. S ) -> ( u i^i S ) =/= (/) ) )
12	9, 11	mpan2d 710	. . . . . . . 8 |- ( ( ph /\ k e. Z ) -> ( ( F ` k ) e. u -> ( u i^i S ) =/= (/) ) )
13	12	adantld 483	. . . . . . 7 |- ( ( ph /\ k e. Z ) -> ( ( k e. dom F /\ ( F ` k ) e. u ) -> ( u i^i S ) =/= (/) ) )
14	13	rexlimdva 3031	. . . . . 6 |- ( ph -> ( E. k e. Z ( k e. dom F /\ ( F ` k ) e. u ) -> ( u i^i S ) =/= (/) ) )
15	8, 14	syl5 34	. . . . 5 |- ( ph -> ( E. j e. Z A. k e. ( ZZ>= ` j ) ( k e. dom F /\ ( F ` k ) e. u ) -> ( u i^i S ) =/= (/) ) )
16	15	imim2d 57	. . . 4 |- ( ph -> ( ( P e. u -> E. j e. Z A. k e. ( ZZ>= ` j ) ( k e. dom F /\ ( F ` k ) e. u ) ) -> ( P e. u -> ( u i^i S ) =/= (/) ) ) )
17	16	ralimdv 2963	. . 3 |- ( ph -> ( A. u e. J ( P e. u -> E. j e. Z A. k e. ( ZZ>= ` j ) ( k e. dom F /\ ( F ` k ) e. u ) ) -> A. u e. J ( P e. u -> ( u i^i S ) =/= (/) ) ) )
18	7, 17	mpd 15	. 2 |- ( ph -> A. u e. J ( P e. u -> ( u i^i S ) =/= (/) ) )
19		topontop 20718	. . . 4 |- ( J e. (TopOn ` X ) -> J e. Top )
20	2, 19	syl 17	. . 3 |- ( ph -> J e. Top )
21		lmcls.8	. . . 4 |- ( ph -> S C_ X )
22		toponuni 20719	. . . . 5 |- ( J e. (TopOn ` X ) -> X = U. J )
23	2, 22	syl 17	. . . 4 |- ( ph -> X = U. J )
24	21, 23	sseqtrd 3641	. . 3 |- ( ph -> S C_ U. J )
25		lmcl 21101	. . . . 5 |- ( ( J e. (TopOn ` X ) /\ F ( ~~> t ` J ) P ) -> P e. X )
26	2, 1, 25	syl2anc 693	. . . 4 |- ( ph -> P e. X )
27	26, 23	eleqtrd 2703	. . 3 |- ( ph -> P e. U. J )
28		eqid 2622	. . . 4 |- U. J = U. J
29	28	elcls 20877	. . 3 |- ( ( J e. Top /\ S C_ U. J /\ P e. U. J ) -> ( P e. ( ( cls ` J ) ` S ) <-> A. u e. J ( P e. u -> ( u i^i S ) =/= (/) ) ) )
30	20, 24, 27, 29	syl3anc 1326	. 2 |- ( ph -> ( P e. ( ( cls ` J ) ` S ) <-> A. u e. J ( P e. u -> ( u i^i S ) =/= (/) ) ) )
31	18, 30	mpbird 247	1 |- ( ph -> P e. ( ( cls ` J ) ` S ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   i^i cin 3573   C_ wss 3574   (/)c0 3915   U.cuni 4436   class class class wbr 4653   dom cdm 5114   ` cfv 5888  (class class class)co 6650   ^pm cpm 7858   CCcc 9934   ZZcz 11377   ZZ>=cuz 11687   Topctop 20698  TopOnctopon 20715   clsccl 20822   ~~> tclm 21030

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  ax-cnex 9992  ax-resscn 9993  ax-pre-lttri 10010  ax-pre-lttrn 10011

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-po 5035  df-so 5036  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-1st 7168  df-2nd 7169  df-er 7742  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-neg 10269  df-z 11378  df-uz 11688  df-top 20699  df-topon 20716  df-cld 20823  df-ntr 20824  df-cls 20825  df-lm 21033

This theorem is referenced by:  lmcld  21107  1stcelcls  21264  caublcls  23107