Theorem neips 7677

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 7673	. . . . . 6 |- ((J e. Top /\ N e. ((nei` J)` S) /\ {p} (_ S) -> N e. ((nei` J)` {p}))
2		snssi 2462	. . . . . 6 |- (p e. S -> {p} (_ S)
3	1, 2	syl3an3 860	. . . . 5 |- ((J e. Top /\ N e. ((nei` J)` S) /\ p e. S) -> N e. ((nei` J)` {p}))
4	3	3exp 831	. . . 4 |- (J e. Top -> (N e. ((nei` J)` S) -> (p e. S -> N e. ((nei` J)` {p}))))
5	4	r19.21adv 1715	. . 3 |- (J e. Top -> (N e. ((nei` J)` S) -> A.p e. S N e. ((nei` J)` {p})))
6	5	3ad2ant1 799	. 2 |- ((J e. Top /\ S (_ X /\ S =/= (/)) -> (N e. ((nei` J)` S) -> A.p e. S N e. ((nei` J)` {p})))
7		r19.28zv 2346	. . . . 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 801	. . . 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 2079	. . . . . . . . . 10 |- (h = U.{v e. J | v (_ N} -> (S (_ h <-> S (_ U.{v e. J | v (_ N}))
10		sseq1 2078	. . . . . . . . . 10 |- (h = U.{v e. J | v (_ N} -> (h (_ N <-> U.{v e. J | v (_ N} (_ N))
11	9, 10	anbi12d 627	. . . . . . . . 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 1873	. . . . . . . 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 2127	. . . . . . . . . 10 |- {v e. J | v (_ N} (_ J
14		uniopnt 7548	. . . . . . . . . 10 |- ((J e. Top /\ {v e. J | v (_ N} (_ J) -> U.{v e. J | v (_ N} e. J)
15	13, 14	mpan2 695	. . . . . . . . 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 2503	. . . . . . . . . . . . . . 15 |- ((p e. g /\ g e. {v e. J | v (_ N}) -> p e. U.{v e. J | v (_ N})
18		sseq1 2078	. . . . . . . . . . . . . . . 16 |- (v = g -> (v (_ N <-> g (_ N))
19	18	elrab 1901	. . . . . . . . . . . . . . 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 1741	. . . . . . . . . . . 12 |- (E.g e. J (p e. g /\ g (_ N) -> p e. U.{v e. J | v (_ N})
23	22	r19.20si 1703	. . . . . . . . . . 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 2055	. . . . . . . . . . 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 2523	. . . . . . . . . 10 |- (U.{v e. J | v (_ N} (_ N <-> A.h e. {v e. J | v (_ N}h (_ N)
28		sseq1 2078	. . . . . . . . . . . 12 |- (v = h -> (v (_ N <-> h (_ N))
29	28	elrab 1901	. . . . . . . . . . 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 1698	. . . . . . . . 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 560	. . . . 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 798	. . . 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 7670	. . . . . . 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 2060	. . . . . . 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 1656	. . . 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 798	. . 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 7668	. . . 4 |- ((J e. Top /\ S (_ X) -> (N e. ((nei` J)` S) <-> (N (_ X /\ E.h e. J (S (_ h /\ h (_ N))))
46	45	3adant3 798	. . 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 542	. 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 515	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 774   = wceq 954   e. wcel 956   =/= wne 1582  A.wral 1642  E.wrex 1643  {crab 1645   (_ wss 2043  (/)c0 2276  {csn 2405  U.cuni 2498  ` cfv 3177  Topctop 7538  neicnei 7662

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-rep 2688  ax-sep 2698  ax-pow 2737  ax-pr 2774  ax-un 2861

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-rab 1649  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-uni 2499  df-br 2615  df-opab 2662  df-id 2830  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188