Theorem innei 7686

Description: The intersection of two neighborhoods of a set is also a neighborhood of the set. Based on Bourbaki TG I.3 Vii. (Contributed by FL, 28-Sep-2006.)

Assertion

Ref	Expression
innei	|- ((J e. Top /\ N e. ((nei` J)` S) /\ M e. ((nei` J)` S)) -> (N i^i M) e. ((nei` J)` S))

Proof of Theorem innei

Step	Hyp	Ref	Expression
1		eqid 1473	. . . . . 6 |- U.J = U.J
2	1	neii1 7671	. . . . 5 |- ((J e. Top /\ N e. ((nei` J)` S)) -> N (_ U.J)
3		ssinss1 2233	. . . . 5 |- (N (_ U.J -> (N i^i M) (_ U.J)
4	2, 3	syl 10	. . . 4 |- ((J e. Top /\ N e. ((nei` J)` S)) -> (N i^i M) (_ U.J)
5	4	3adant3 798	. . 3 |- ((J e. Top /\ N e. ((nei` J)` S) /\ M e. ((nei` J)` S)) -> (N i^i M) (_ U.J)
6		neii2 7672	. . . . . . 7 |- ((J e. Top /\ N e. ((nei` J)` S)) -> E.h e. J (S (_ h /\ h (_ N))
7		neii2 7672	. . . . . . 7 |- ((J e. Top /\ M e. ((nei` J)` S)) -> E.v e. J (S (_ v /\ v (_ M))
8	6, 7	anim12i 333	. . . . . 6 |- (((J e. Top /\ N e. ((nei` J)` S)) /\ (J e. Top /\ M e. ((nei` J)` S))) -> (E.h e. J (S (_ h /\ h (_ N) /\ E.v e. J (S (_ v /\ v (_ M)))
9	8	anandis 512	. . . . 5 |- ((J e. Top /\ (N e. ((nei` J)` S) /\ M e. ((nei` J)` S))) -> (E.h e. J (S (_ h /\ h (_ N) /\ E.v e. J (S (_ v /\ v (_ M)))
10		sseq2 2079	. . . . . . . . . . . . . . 15 |- (g = (h i^i v) -> (S (_ g <-> S (_ (h i^i v)))
11		sseq1 2078	. . . . . . . . . . . . . . 15 |- (g = (h i^i v) -> (g (_ (N i^i M) <-> (h i^i v) (_ (N i^i M)))
12	10, 11	anbi12d 627	. . . . . . . . . . . . . 14 |- (g = (h i^i v) -> ((S (_ g /\ g (_ (N i^i M)) <-> (S (_ (h i^i v) /\ (h i^i v) (_ (N i^i M))))
13	12	rcla4ev 1873	. . . . . . . . . . . . 13 |- (((h i^i v) e. J /\ (S (_ (h i^i v) /\ (h i^i v) (_ (N i^i M))) -> E.g e. J (S (_ g /\ g (_ (N i^i M)))
14		inopnt 7550	. . . . . . . . . . . . . 14 |- ((J e. Top /\ h e. J /\ v e. J) -> (h i^i v) e. J)
15	14	3expa 832	. . . . . . . . . . . . 13 |- (((J e. Top /\ h e. J) /\ v e. J) -> (h i^i v) e. J)
16		ssin 2228	. . . . . . . . . . . . . . . 16 |- ((S (_ h /\ S (_ v) <-> S (_ (h i^i v))
17	16	biimp 151	. . . . . . . . . . . . . . 15 |- ((S (_ h /\ S (_ v) -> S (_ (h i^i v))
18		ss2in 2232	. . . . . . . . . . . . . . 15 |- ((h (_ N /\ v (_ M) -> (h i^i v) (_ (N i^i M))
19	17, 18	anim12i 333	. . . . . . . . . . . . . 14 |- (((S (_ h /\ S (_ v) /\ (h (_ N /\ v (_ M)) -> (S (_ (h i^i v) /\ (h i^i v) (_ (N i^i M)))
20	19	an4s 508	. . . . . . . . . . . . 13 |- (((S (_ h /\ h (_ N) /\ (S (_ v /\ v (_ M)) -> (S (_ (h i^i v) /\ (h i^i v) (_ (N i^i M)))
21	13, 15, 20	syl2an 454	. . . . . . . . . . . 12 |- ((((J e. Top /\ h e. J) /\ v e. J) /\ ((S (_ h /\ h (_ N) /\ (S (_ v /\ v (_ M))) -> E.g e. J (S (_ g /\ g (_ (N i^i M)))
22	21	anassrs 441	. . . . . . . . . . 11 |- (((((J e. Top /\ h e. J) /\ v e. J) /\ (S (_ h /\ h (_ N)) /\ (S (_ v /\ v (_ M)) -> E.g e. J (S (_ g /\ g (_ (N i^i M)))
23	22	ex 373	. . . . . . . . . 10 |- ((((J e. Top /\ h e. J) /\ v e. J) /\ (S (_ h /\ h (_ N)) -> ((S (_ v /\ v (_ M) -> E.g e. J (S (_ g /\ g (_ (N i^i M))))
24	23	an1rs 489	. . . . . . . . 9 |- ((((J e. Top /\ h e. J) /\ (S (_ h /\ h (_ N)) /\ v e. J) -> ((S (_ v /\ v (_ M) -> E.g e. J (S (_ g /\ g (_ (N i^i M))))
25	24	r19.23adva 1744	. . . . . . . 8 |- (((J e. Top /\ h e. J) /\ (S (_ h /\ h (_ N)) -> (E.v e. J (S (_ v /\ v (_ M) -> E.g e. J (S (_ g /\ g (_ (N i^i M))))
26	25	ex 373	. . . . . . 7 |- ((J e. Top /\ h e. J) -> ((S (_ h /\ h (_ N) -> (E.v e. J (S (_ v /\ v (_ M) -> E.g e. J (S (_ g /\ g (_ (N i^i M)))))
27	26	r19.23adva 1744	. . . . . 6 |- (J e. Top -> (E.h e. J (S (_ h /\ h (_ N) -> (E.v e. J (S (_ v /\ v (_ M) -> E.g e. J (S (_ g /\ g (_ (N i^i M)))))
28	27	imp32 363	. . . . 5 |- ((J e. Top /\ (E.h e. J (S (_ h /\ h (_ N) /\ E.v e. J (S (_ v /\ v (_ M))) -> E.g e. J (S (_ g /\ g (_ (N i^i M)))
29	9, 28	syldan 467	. . . 4 |- ((J e. Top /\ (N e. ((nei` J)` S) /\ M e. ((nei` J)` S))) -> E.g e. J (S (_ g /\ g (_ (N i^i M)))
30	29	3impb 828	. . 3 |- ((J e. Top /\ N e. ((nei` J)` S) /\ M e. ((nei` J)` S)) -> E.g e. J (S (_ g /\ g (_ (N i^i M)))
31	5, 30	jca 288	. 2 |- ((J e. Top /\ N e. ((nei` J)` S) /\ M e. ((nei` J)` S)) -> ((N i^i M) (_ U.J /\ E.g e. J (S (_ g /\ g (_ (N i^i M))))
32	1	neiss2 7666	. . . 4 |- ((J e. Top /\ N e. ((nei` J)` S)) -> S (_ U.J)
33	1	isnei 7668	. . . 4 |- ((J e. Top /\ S (_ U.J) -> ((N i^i M) e. ((nei` J)` S) <-> ((N i^i M) (_ U.J /\ E.g e. J (S (_ g /\ g (_ (N i^i M)))))
34	32, 33	syldan 467	. . 3 |- ((J e. Top /\ N e. ((nei` J)` S)) -> ((N i^i M) e. ((nei` J)` S) <-> ((N i^i M) (_ U.J /\ E.g e. J (S (_ g /\ g (_ (N i^i M)))))
35	34	3adant3 798	. 2 |- ((J e. Top /\ N e. ((nei` J)` S) /\ M e. ((nei` J)` S)) -> ((N i^i M) e. ((nei` J)` S) <-> ((N i^i M) (_ U.J /\ E.g e. J (S (_ g /\ g (_ (N i^i M)))))
36	31, 35	mpbird 196	1 |- ((J e. Top /\ N e. ((nei` J)` S) /\ M e. ((nei` J)` S)) -> (N i^i M) e. ((nei` J)` S))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 774   = wceq 954   e. wcel 956  E.wrex 1643   i^i cin 2042   (_ wss 2043  U.cuni 2498  ` cfv 3177  Topctop 7538  neicnei 7662

This theorem is referenced by:  neifil 10478

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  df-fv 3193  df-top 7542  df-nei 7663

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