Theorem neips 7727

Description: A neighborhood of a set is a neighborhood of every point in the set. Bourbaki TG I.2. (Contributed by FL, 16-Nov-2006.)

Hypothesis

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

Assertion

Ref	Expression
neips	|- ((J e. Top /\ S (_ X /\ S =/= (/)) -> (N e. ((nei` J)` S) <-> A.p e. S N e. ((nei` J)` {p})))

Distinct variable groups:   J,p   N,p   S,p   X,p

Proof of Theorem neips

Step	Hyp	Ref	Expression
1		neiss 7723	. . . . . 6 |- ((J e. Top /\ N e. ((nei` J)` S) /\ {p} (_ S) -> N e. ((nei` J)` {p}))
2		snssi 2466	. . . . . 6 |- (p e. S -> {p} (_ S)
3	1, 2	syl3an3 861	. . . . 5 |- ((J e. Top /\ N e. ((nei` J)` S) /\ p e. S) -> N e. ((nei` J)` {p}))
4	3	3exp 832	. . . 4 |- (J e. Top -> (N e. ((nei` J)` S) -> (p e. S -> N e. ((nei` J)` {p}))))
5	4	r19.21adv 1718	. . 3 |- (J e. Top -> (N e. ((nei` J)` S) -> A.p e. S N e. ((nei` J)` {p})))
6	5	3ad2ant1 800	. 2 |- ((J e. Top /\ S (_ X /\ S =/= (/)) -> (N e. ((nei` J)` S) -> A.p e. S N e. ((nei` J)` {p})))
7		r19.28zv 2350	. . . . 5 |- (S =/= (/) -> (A.p e. S (N (_ X /\ E.g e. J (p e. g /\ g (_ N)) <-> (N (_ X /\ A.p e. S E.g e. J (p e. g /\ g (_ N))))
8	7	3ad2ant3 802	. . . 4 |- ((J e. Top /\ S (_ X /\ S =/= (/)) -> (A.p e. S (N (_ X /\ E.g e. J (p e. g /\ g (_ N)) <-> (N (_ X /\ A.p e. S E.g e. J (p e. g /\ g (_ N))))
9		sseq2 2083	. . . . . . . . . 10 |- (h = U.{v e. J | v (_ N} -> (S (_ h <-> S (_ U.{v e. J | v (_ N}))
10		sseq1 2082	. . . . . . . . . 10 |- (h = U.{v e. J | v (_ N} -> (h (_ N <-> U.{v e. J | v (_ N} (_ N))
11	9, 10	anbi12d 628	. . . . . . . . 9 |- (h = U.{v e. J | v (_ N} -> ((S (_ h /\ h (_ N) <-> (S (_ U.{v e. J | v (_ N} /\ U.{v e. J | v (_ N} (_ N)))
12	11	rcla4ev 1877	. . . . . . . 8 |- ((U.{v e. J | v (_ N} e. J /\ (S (_ U.{v e. J | v (_ N} /\ U.{v e. J | v (_ N} (_ N)) -> E.h e. J (S (_ h /\ h (_ N))
13		ssrab2 2131	. . . . . . . . . 10 |- {v e. J | v (_ N} (_ J
14		uniopnt 7598	. . . . . . . . . 10 |- ((J e. Top /\ {v e. J | v (_ N} (_ J) -> U.{v e. J | v (_ N} e. J)
15	13, 14	mpan2 696	. . . . . . . . 9 |- (J e. Top -> U.{v e. J | v (_ N} e. J)
16	15	ad2antrr 404	. . . . . . . 8 |- (((J e. Top /\ S (_ X) /\ A.p e. S E.g e. J (p e. g /\ g (_ N)) -> U.{v e. J | v (_ N} e. J)
17		elunii 2508	. . . . . . . . . . . . . . 15 |- ((p e. g /\ g e. {v e. J | v (_ N}) -> p e. U.{v e. J | v (_ N})
18		sseq1 2082	. . . . . . . . . . . . . . . 16 |- (v = g -> (v (_ N <-> g (_ N))
19	18	elrab 1905	. . . . . . . . . . . . . . 15 |- (g e. {v e. J | v (_ N} <-> (g e. J /\ g (_ N))
20	17, 19	sylan2br 453	. . . . . . . . . . . . . 14 |- ((p e. g /\ (g e. J /\ g (_ N)) -> p e. U.{v e. J | v (_ N})
21	20	an1s 486	. . . . . . . . . . . . 13 |- ((g e. J /\ (p e. g /\ g (_ N)) -> p e. U.{v e. J | v (_ N})
22	21	r19.23aiva 1744	. . . . . . . . . . . 12 |- (E.g e. J (p e. g /\ g (_ N) -> p e. U.{v e. J | v (_ N})
23	22	r19.20si 1706	. . . . . . . . . . 11 |- (A.p e. S E.g e. J (p e. g /\ g (_ N) -> A.p e. S p e. U.{v e. J | v (_ N})
24		dfss3 2059	. . . . . . . . . . 11 |- (S (_ U.{v e. J | v (_ N} <-> A.p e. S p e. U.{v e. J | v (_ N})
25	23, 24	sylibr 200	. . . . . . . . . 10 |- (A.p e. S E.g e. J (p e. g /\ g (_ N) -> S (_ U.{v e. J | v (_ N})
26	25	adantl 388	. . . . . . . . 9 |- (((J e. Top /\ S (_ X) /\ A.p e. S E.g e. J (p e. g /\ g (_ N)) -> S (_ U.{v e. J | v (_ N})
27		unissb 2528	. . . . . . . . . 10 |- (U.{v e. J | v (_ N} (_ N <-> A.h e. {v e. J | v (_ N}h (_ N)
28		sseq1 2082	. . . . . . . . . . . 12 |- (v = h -> (v (_ N <-> h (_ N))
29	28	elrab 1905	. . . . . . . . . . 11 |- (h e. {v e. J | v (_ N} <-> (h e. J /\ h (_ N))
30	29	pm3.27bi 326	. . . . . . . . . 10 |- (h e. {v e. J | v (_ N} -> h (_ N)
31	27, 30	mprgbir 1701	. . . . . . . . 9 |- U.{v e. J | v (_ N} (_ N
32	26, 31	jctir 293	. . . . . . . 8 |- (((J e. Top /\ S (_ X) /\ A.p e. S E.g e. J (p e. g /\ g (_ N)) -> (S (_ U.{v e. J | v (_ N} /\ U.{v e. J | v (_ N} (_ N))
33	12, 16, 32	sylanc 471	. . . . . . 7 |- (((J e. Top /\ S (_ X) /\ A.p e. S E.g e. J (p e. g /\ g (_ N)) -> E.h e. J (S (_ h /\ h (_ N))
34	33	ex 373	. . . . . 6 |- ((J e. Top /\ S (_ X) -> (A.p e. S E.g e. J (p e. g /\ g (_ N) -> E.h e. J (S (_ h /\ h (_ N)))
35	34	anim2d 561	. . . . 5 |- ((J e. Top /\ S (_ X) -> ((N (_ X /\ A.p e. S E.g e. J (p e. g /\ g (_ N)) -> (N (_ X /\ E.h e. J (S (_ h /\ h (_ N))))
36	35	3adant3 799	. . . 4 |- ((J e. Top /\ S (_ X /\ S =/= (/)) -> ((N (_ X /\ A.p e. S E.g e. J (p e. g /\ g (_ N)) -> (N (_ X /\ E.h e. J (S (_ h /\ h (_ N))))
37	8, 36	sylbid 203	. . 3 |- ((J e. Top /\ S (_ X /\ S =/= (/)) -> (A.p e. S (N (_ X /\ E.g e. J (p e. g /\ g (_ N)) -> (N (_ X /\ E.h e. J (S (_ h /\ h (_ N))))
38		neips.1	. . . . . . . 8 |- X = U.J
39	38	isneip 7720	. . . . . . 7 |- ((J e. Top /\ p e. X) -> (N e. ((nei` J)` {p}) <-> (N (_ X /\ E.g e. J (p e. g /\ g (_ N))))
40		ssel2 2064	. . . . . . 7 |- ((S (_ X /\ p e. S) -> p e. X)
41	39, 40	sylan2 451	. . . . . 6 |- ((J e. Top /\ (S (_ X /\ p e. S)) -> (N e. ((nei` J)` {p}) <-> (N (_ X /\ E.g e. J (p e. g /\ g (_ N))))
42	41	anassrs 441	. . . . 5 |- (((J e. Top /\ S (_ X) /\ p e. S) -> (N e. ((nei` J)` {p}) <-> (N (_ X /\ E.g e. J (p e. g /\ g (_ N))))
43	42	ralbidva 1659	. . . 4 |- ((J e. Top /\ S (_ X) -> (A.p e. S N e. ((nei` J)` {p}) <-> A.p e. S (N (_ X /\ E.g e. J (p e. g /\ g (_ N))))
44	43	3adant3 799	. . 3 |- ((J e. Top /\ S (_ X /\ S =/= (/)) -> (A.p e. S N e. ((nei` J)` {p}) <-> A.p e. S (N (_ X /\ E.g e. J (p e. g /\ g (_ N))))
45	38	isnei 7718	. . . 4 |- ((J e. Top /\ S (_ X) -> (N e. ((nei` J)` S) <-> (N (_ X /\ E.h e. J (S (_ h /\ h (_ N))))
46	45	3adant3 799	. . 3 |- ((J e. Top /\ S (_ X /\ S =/= (/)) -> (N e. ((nei` J)` S) <-> (N (_ X /\ E.h e. J (S (_ h /\ h (_ N))))
47	37, 44, 46	3imtr4d 543	. 2 |- ((J e. Top /\ S (_ X /\ S =/= (/)) -> (A.p e. S N e. ((nei` J)` {p}) -> N e. ((nei` J)` S)))
48	6, 47	impbid 516	1 |- ((J e. Top /\ S (_ X /\ S =/= (/)) -> (N e. ((nei` J)` S) <-> A.p e. S N e. ((nei` J)` {p})))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 775   = wceq 956   e. wcel 958   =/= wne 1585  A.wral 1645  E.wrex 1646  {crab 1648   (_ wss 2047  (/)c0 2280  {csn 2409  U.cuni 2503  ` cfv 3182  Topctop 7588  neicnei 7712

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-rab 1652  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192