Theorem fclusfnei 15626

Description: The property of being a cluster point of a function in terms of neighborhoods.

Hypotheses

Ref	Expression
fclusfnei.1	|- X = U.J
fclusfnei.2	|- Y = U.L

Assertion

Ref	Expression
fclusfnei	|- ((J e. Top /\ L e. Fil /\ 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)) =/= (/))))

Distinct variable groups:   n,s,A   n,F,s   n,J,s   n,L,s   n,X,s   n,Y,s

Proof of Theorem fclusfnei

Step	Hyp	Ref	Expression
1		fclusfnei.1	. . 3 |- X = U.J
2		fclusfnei.2	. . 3 |- Y = U.L
3	1, 2	isfclusf 15625	. 2 |- ((J e. Top /\ L e. Fil /\ F:Y-->X) -> (A e. ((J fClusf L)` F) <-> (A e. X /\ A.o e. J (A e. o -> A.s e. L (o i^i (F"s)) =/= (/)))))
4		neii2 8998	. . . . . . . . . 10 |- ((J e. Top /\ n e. ((nei` J)` {A})) -> E.j e. J ({A} C_ j /\ j C_ n))
5	4	ex 402	. . . . . . . . 9 |- (J e. Top -> (n e. ((nei` J)` {A}) -> E.j e. J ({A} C_ j /\ j C_ n)))
6	5	3ad2ant1 897	. . . . . . . 8 |- ((J e. Top /\ L e. Fil /\ F:Y-->X) -> (n e. ((nei` J)` {A}) -> E.j e. J ({A} C_ j /\ j C_ n)))
7	6	adantr 425	. . . . . . 7 |- (((J e. Top /\ L e. Fil /\ F:Y-->X) /\ A e. X) -> (n e. ((nei` J)` {A}) -> E.j e. J ({A} C_ j /\ j C_ n)))
8		snssg 3124	. . . . . . . . . . . . . 14 |- (A e. X -> (A e. j <-> {A} C_ j))
9	8	biimpar 461	. . . . . . . . . . . . 13 |- ((A e. X /\ {A} C_ j) -> A e. j)
10	9	adantrr 431	. . . . . . . . . . . 12 |- ((A e. X /\ ({A} C_ j /\ j C_ n)) -> A e. j)
11	10	ad2ant2l 444	. . . . . . . . . . 11 |- ((((J e. Top /\ L e. Fil /\ F:Y-->X) /\ A e. X) /\ (j e. J /\ ({A} C_ j /\ j C_ n))) -> A e. j)
12		eleq2 1958	. . . . . . . . . . . . . 14 |- (o = j -> (A e. o <-> A e. j))
13		ineq1 2789	. . . . . . . . . . . . . . . 16 |- (o = j -> (o i^i (F"s)) = (j i^i (F"s)))
14	13	neeq1d 2028	. . . . . . . . . . . . . . 15 |- (o = j -> ((o i^i (F"s)) =/= (/) <-> (j i^i (F"s)) =/= (/)))
15	14	ralbidv 2123	. . . . . . . . . . . . . 14 |- (o = j -> (A.s e. L (o i^i (F"s)) =/= (/) <-> A.s e. L (j i^i (F"s)) =/= (/)))
16	12, 15	imbi12d 688	. . . . . . . . . . . . 13 |- (o = j -> ((A e. o -> A.s e. L (o i^i (F"s)) =/= (/)) <-> (A e. j -> A.s e. L (j i^i (F"s)) =/= (/))))
17	16	rcla4v 2376	. . . . . . . . . . . 12 |- (j e. J -> (A.o e. J (A e. o -> A.s e. L (o i^i (F"s)) =/= (/)) -> (A e. j -> A.s e. L (j i^i (F"s)) =/= (/))))
18	17	ad2antrl 442	. . . . . . . . . . 11 |- ((((J e. Top /\ L e. Fil /\ F:Y-->X) /\ A e. X) /\ (j e. J /\ ({A} C_ j /\ j C_ n))) -> (A.o e. J (A e. o -> A.s e. L (o i^i (F"s)) =/= (/)) -> (A e. j -> A.s e. L (j i^i (F"s)) =/= (/))))
19	11, 18	mpid 58	. . . . . . . . . 10 |- ((((J e. Top /\ L e. Fil /\ F:Y-->X) /\ A e. X) /\ (j e. J /\ ({A} C_ j /\ j C_ n))) -> (A.o e. J (A e. o -> A.s e. L (o i^i (F"s)) =/= (/)) -> A.s e. L (j i^i (F"s)) =/= (/)))
20		ssrin 2817	. . . . . . . . . . . . . 14 |- (j C_ n -> (j i^i (F"s)) C_ (n i^i (F"s)))
21		ssn0 2905	. . . . . . . . . . . . . . 15 |- (((j i^i (F"s)) C_ (n i^i (F"s)) /\ (j i^i (F"s)) =/= (/)) -> (n i^i (F"s)) =/= (/))
22	21	ex 402	. . . . . . . . . . . . . 14 |- ((j i^i (F"s)) C_ (n i^i (F"s)) -> ((j i^i (F"s)) =/= (/) -> (n i^i (F"s)) =/= (/)))
23	20, 22	syl 12	. . . . . . . . . . . . 13 |- (j C_ n -> ((j i^i (F"s)) =/= (/) -> (n i^i (F"s)) =/= (/)))
24	23	ralimdv 2172	. . . . . . . . . . . 12 |- (j C_ n -> (A.s e. L (j i^i (F"s)) =/= (/) -> A.s e. L (n i^i (F"s)) =/= (/)))
25	24	adantl 424	. . . . . . . . . . 11 |- (({A} C_ j /\ j C_ n) -> (A.s e. L (j i^i (F"s)) =/= (/) -> A.s e. L (n i^i (F"s)) =/= (/)))
26	25	ad2antll 443	. . . . . . . . . 10 |- ((((J e. Top /\ L e. Fil /\ F:Y-->X) /\ A e. X) /\ (j e. J /\ ({A} C_ j /\ j C_ n))) -> (A.s e. L (j i^i (F"s)) =/= (/) -> A.s e. L (n i^i (F"s)) =/= (/)))
27	19, 26	syld 30	. . . . . . . . 9 |- ((((J e. Top /\ L e. Fil /\ F:Y-->X) /\ A e. X) /\ (j e. J /\ ({A} C_ j /\ j C_ n))) -> (A.o e. J (A e. o -> A.s e. L (o i^i (F"s)) =/= (/)) -> A.s e. L (n i^i (F"s)) =/= (/)))
28	27	exp32 408	. . . . . . . 8 |- (((J e. Top /\ L e. Fil /\ F:Y-->X) /\ A e. X) -> (j e. J -> (({A} C_ j /\ j C_ n) -> (A.o e. J (A e. o -> A.s e. L (o i^i (F"s)) =/= (/)) -> A.s e. L (n i^i (F"s)) =/= (/)))))
29	28	r19.23adv 2215	. . . . . . 7 |- (((J e. Top /\ L e. Fil /\ F:Y-->X) /\ A e. X) -> (E.j e. J ({A} C_ j /\ j C_ n) -> (A.o e. J (A e. o -> A.s e. L (o i^i (F"s)) =/= (/)) -> A.s e. L (n i^i (F"s)) =/= (/))))
30	7, 29	syld 30	. . . . . 6 |- (((J e. Top /\ L e. Fil /\ F:Y-->X) /\ A e. X) -> (n e. ((nei` J)` {A}) -> (A.o e. J (A e. o -> A.s e. L (o i^i (F"s)) =/= (/)) -> A.s e. L (n i^i (F"s)) =/= (/))))
31	30	com23 36	. . . . 5 |- (((J e. Top /\ L e. Fil /\ F:Y-->X) /\ A e. X) -> (A.o e. J (A e. o -> A.s e. L (o i^i (F"s)) =/= (/)) -> (n e. ((nei` J)` {A}) -> A.s e. L (n i^i (F"s)) =/= (/))))
32	31	r19.21adv 2181	. . . 4 |- (((J e. Top /\ L e. Fil /\ F:Y-->X) /\ A e. X) -> (A.o e. J (A e. o -> A.s e. L (o i^i (F"s)) =/= (/)) -> A.n e. ((nei` J)` {A})A.s e. L (n i^i (F"s)) =/= (/)))
33		opnneip 9009	. . . . . . . . . . . 12 |- ((J e. Top /\ o e. J /\ A e. o) -> o e. ((nei` J)` {A}))
34	33	3expb 1068	. . . . . . . . . . 11 |- ((J e. Top /\ (o e. J /\ A e. o)) -> o e. ((nei` J)` {A}))
35	34	3ad2antl1 1038	. . . . . . . . . 10 |- (((J e. Top /\ L e. Fil /\ F:Y-->X) /\ (o e. J /\ A e. o)) -> o e. ((nei` J)` {A}))
36	35	adantrl 430	. . . . . . . . 9 |- (((J e. Top /\ L e. Fil /\ F:Y-->X) /\ (A e. X /\ (o e. J /\ A e. o))) -> o e. ((nei` J)` {A}))
37		ineq1 2789	. . . . . . . . . . . 12 |- (n = o -> (n i^i (F"s)) = (o i^i (F"s)))
38	37	neeq1d 2028	. . . . . . . . . . 11 |- (n = o -> ((n i^i (F"s)) =/= (/) <-> (o i^i (F"s)) =/= (/)))
39	38	ralbidv 2123	. . . . . . . . . 10 |- (n = o -> (A.s e. L (n i^i (F"s)) =/= (/) <-> A.s e. L (o i^i (F"s)) =/= (/)))
40	39	rcla4v 2376	. . . . . . . . 9 |- (o e. ((nei` J)` {A}) -> (A.n e. ((nei` J)` {A})A.s e. L (n i^i (F"s)) =/= (/) -> A.s e. L (o i^i (F"s)) =/= (/)))
41	36, 40	syl 12	. . . . . . . 8 |- (((J e. Top /\ L e. Fil /\ F:Y-->X) /\ (A e. X /\ (o e. J /\ A e. o))) -> (A.n e. ((nei` J)` {A})A.s e. L (n i^i (F"s)) =/= (/) -> A.s e. L (o i^i (F"s)) =/= (/)))
42	41	expr 418	. . . . . . 7 |- (((J e. Top /\ L e. Fil /\ F:Y-->X) /\ A e. X) -> ((o e. J /\ A e. o) -> (A.n e. ((nei` J)` {A})A.s e. L (n i^i (F"s)) =/= (/) -> A.s e. L (o i^i (F"s)) =/= (/))))
43	42	com23 36	. . . . . 6 |- (((J e. Top /\ L e. Fil /\ F:Y-->X) /\ A e. X) -> (A.n e. ((nei` J)` {A})A.s e. L (n i^i (F"s)) =/= (/) -> ((o e. J /\ A e. o) -> A.s e. L (o i^i (F"s)) =/= (/))))
44	43	exp4a 409	. . . . 5 |- (((J e. Top /\ L e. Fil /\ F:Y-->X) /\ A e. X) -> (A.n e. ((nei` J)` {A})A.s e. L (n i^i (F"s)) =/= (/) -> (o e. J -> (A e. o -> A.s e. L (o i^i (F"s)) =/= (/)))))
45	44	r19.21adv 2181	. . . 4 |- (((J e. Top /\ L e. Fil /\ F:Y-->X) /\ A e. X) -> (A.n e. ((nei` J)` {A})A.s e. L (n i^i (F"s)) =/= (/) -> A.o e. J (A e. o -> A.s e. L (o i^i (F"s)) =/= (/))))
46	32, 45	impbid 574	. . 3 |- (((J e. Top /\ L e. Fil /\ F:Y-->X) /\ A e. X) -> (A.o e. J (A e. o -> A.s e. L (o i^i (F"s)) =/= (/)) <-> A.n e. ((nei` J)` {A})A.s e. L (n i^i (F"s)) =/= (/)))
47	46	pm5.32da 711	. 2 |- ((J e. Top /\ L e. Fil /\ F:Y-->X) -> ((A e. X /\ A.o e. J (A e. o -> A.s e. L (o i^i (F"s)) =/= (/))) <-> (A e. X /\ A.n e. ((nei` J)` {A})A.s e. L (n i^i (F"s)) =/= (/))))
48	3, 47	bitrd 587	1 |- ((J e. Top /\ L e. Fil /\ 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)) =/= (/))))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300   =/= wne 2017  A.wral 2105  E.wrex 2106   i^i cin 2592   C_ wss 2593  (/)c0 2875  {csn 3044  U.cuni 3177  "cima 3989  -->wf 3994  ` cfv 3998  (class class class)co 4884  Topctop 8857  neicnei 8988  Filcfil 10264   fClusf cfclusf 15583

This theorem is referenced by:  fclsfneii 15628

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-nel 2020  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-iin 3258  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-fv 4014  df-opr 4886  df-oprab 4887  df-map 5383  df-top 8861  df-cld 8939  df-ntr 8940  df-cls 8941  df-nei 8989  df-fbas 10259  df-fg 10260  df-fil 10265  df-filmap 10306  df-fclus 15584  df-fclusf 15585

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