Theorem innei 12015

Description: The intersection of two neighborhoods of a set is also a neighborhood of the set. Generalization to subsets of Property V_ii of [BourbakiTop1] p. I.3 for binary intersections. (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

Dummy variables g h v are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2095	. . . . 5 |- U. J = U. J
2	1	neii1 11999	. . . 4 |- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) ) -> N C_ U. J )
3		ssinss1 3244	. . . 4 |- ( N C_ U. J -> ( N i^i M ) C_ U. J )
4	2, 3	syl 14	. . 3 |- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) ) -> ( N i^i M ) C_ U. J )
5	4	3adant3 966	. 2 |- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) /\ M e. ( ( nei ` J ) ` S ) ) -> ( N i^i M ) C_ U. J )
6		neii2 12001	. . . . 5 |- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) ) -> E. h e. J ( S C_ h /\ h C_ N ) )
7		neii2 12001	. . . . 5 |- ( ( J e. Top /\ M e. ( ( nei ` J ) ` S ) ) -> E. v e. J ( S C_ v /\ v C_ M ) )
8	6, 7	anim12dan 568	. . . 4 |- ( ( J e. Top /\ ( N e. ( ( nei ` J ) ` S ) /\ M e. ( ( nei ` J ) ` S ) ) ) -> ( E. h e. J ( S C_ h /\ h C_ N ) /\ E. v e. J ( S C_ v /\ v C_ M ) ) )
9		inopn 11854	. . . . . . . . . . 11 |- ( ( J e. Top /\ h e. J /\ v e. J ) -> ( h i^i v ) e. J )
10	9	3expa 1146	. . . . . . . . . 10 |- ( ( ( J e. Top /\ h e. J ) /\ v e. J ) -> ( h i^i v ) e. J )
11		ssin 3237	. . . . . . . . . . . . 13 |- ( ( S C_ h /\ S C_ v ) <-> S C_ ( h i^i v ) )
12	11	biimpi 119	. . . . . . . . . . . 12 |- ( ( S C_ h /\ S C_ v ) -> S C_ ( h i^i v ) )
13		ss2in 3243	. . . . . . . . . . . 12 |- ( ( h C_ N /\ v C_ M ) -> ( h i^i v ) C_ ( N i^i M ) )
14	12, 13	anim12i 332	. . . . . . . . . . 11 |- ( ( ( S C_ h /\ S C_ v ) /\ ( h C_ N /\ v C_ M ) ) -> ( S C_ ( h i^i v ) /\ ( h i^i v ) C_ ( N i^i M ) ) )
15	14	an4s 556	. . . . . . . . . 10 |- ( ( ( S C_ h /\ h C_ N ) /\ ( S C_ v /\ v C_ M ) ) -> ( S C_ ( h i^i v ) /\ ( h i^i v ) C_ ( N i^i M ) ) )
16		sseq2 3063	. . . . . . . . . . . 12 |- ( g = ( h i^i v ) -> ( S C_ g <-> S C_ ( h i^i v ) ) )
17		sseq1 3062	. . . . . . . . . . . 12 |- ( g = ( h i^i v ) -> ( g C_ ( N i^i M ) <-> ( h i^i v ) C_ ( N i^i M ) ) )
18	16, 17	anbi12d 458	. . . . . . . . . . 11 |- ( g = ( h i^i v ) -> ( ( S C_ g /\ g C_ ( N i^i M ) ) <-> ( S C_ ( h i^i v ) /\ ( h i^i v ) C_ ( N i^i M ) ) ) )
19	18	rspcev 2736	. . . . . . . . . 10 |- ( ( ( h i^i v ) e. J /\ ( S C_ ( h i^i v ) /\ ( h i^i v ) C_ ( N i^i M ) ) ) -> E. g e. J ( S C_ g /\ g C_ ( N i^i M ) ) )
20	10, 15, 19	syl2an 284	. . . . . . . . 9 |- ( ( ( ( J e. Top /\ h e. J ) /\ v e. J ) /\ ( ( S C_ h /\ h C_ N ) /\ ( S C_ v /\ v C_ M ) ) ) -> E. g e. J ( S C_ g /\ g C_ ( N i^i M ) ) )
21	20	expr 368	. . . . . . . 8 |- ( ( ( ( J e. Top /\ h e. J ) /\ v e. J ) /\ ( S C_ h /\ h C_ N ) ) -> ( ( S C_ v /\ v C_ M ) -> E. g e. J ( S C_ g /\ g C_ ( N i^i M ) ) ) )
22	21	an32s 536	. . . . . . 7 |- ( ( ( ( J e. Top /\ h e. J ) /\ ( S C_ h /\ h C_ N ) ) /\ v e. J ) -> ( ( S C_ v /\ v C_ M ) -> E. g e. J ( S C_ g /\ g C_ ( N i^i M ) ) ) )
23	22	rexlimdva 2502	. . . . . 6 |- ( ( ( J e. Top /\ h e. J ) /\ ( S C_ h /\ h C_ N ) ) -> ( E. v e. J ( S C_ v /\ v C_ M ) -> E. g e. J ( S C_ g /\ g C_ ( N i^i M ) ) ) )
24	23	rexlimdva2 2505	. . . . 5 |- ( J e. Top -> ( E. h e. J ( S C_ h /\ h C_ N ) -> ( E. v e. J ( S C_ v /\ v C_ M ) -> E. g e. J ( S C_ g /\ g C_ ( N i^i M ) ) ) ) )
25	24	imp32 254	. . . 4 |- ( ( J e. Top /\ ( E. h e. J ( S C_ h /\ h C_ N ) /\ E. v e. J ( S C_ v /\ v C_ M ) ) ) -> E. g e. J ( S C_ g /\ g C_ ( N i^i M ) ) )
26	8, 25	syldan 277	. . 3 |- ( ( J e. Top /\ ( N e. ( ( nei ` J ) ` S ) /\ M e. ( ( nei ` J ) ` S ) ) ) -> E. g e. J ( S C_ g /\ g C_ ( N i^i M ) ) )
27	26	3impb 1142	. 2 |- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) /\ M e. ( ( nei ` J ) ` S ) ) -> E. g e. J ( S C_ g /\ g C_ ( N i^i M ) ) )
28	1	neiss2 11994	. . . 4 |- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) ) -> S C_ U. J )
29	1	isnei 11996	. . . 4 |- ( ( J e. Top /\ S C_ U. J ) -> ( ( N i^i M ) e. ( ( nei ` J ) ` S ) <-> ( ( N i^i M ) C_ U. J /\ E. g e. J ( S C_ g /\ g C_ ( N i^i M ) ) ) ) )
30	28, 29	syldan 277	. . 3 |- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) ) -> ( ( N i^i M ) e. ( ( nei ` J ) ` S ) <-> ( ( N i^i M ) C_ U. J /\ E. g e. J ( S C_ g /\ g C_ ( N i^i M ) ) ) ) )
31	30	3adant3 966	. 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 ) C_ U. J /\ E. g e. J ( S C_ g /\ g C_ ( N i^i M ) ) ) ) )
32	5, 27, 31	mpbir2and 893	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 4   /\ wa 103   <-> wb 104   /\ w3a 927   = wceq 1296   e. wcel 1445   E.wrex 2371   i^i cin 3012   C_ wss 3013   U.cuni 3675   ` cfv 5049   Topctop 11848   neicnei 11990

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-coll 3975  ax-sep 3978  ax-pow 4030  ax-pr 4060

This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-reu 2377  df-rab 2379  df-v 2635  df-sbc 2855  df-csb 2948  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-iun 3754  df-br 3868  df-opab 3922  df-mpt 3923  df-id 4144  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-res 4479  df-ima 4480  df-iota 5014  df-fun 5051  df-fn 5052  df-f 5053  df-f1 5054  df-fo 5055  df-f1o 5056  df-fv 5057  df-top 11849  df-nei 11991

This theorem is referenced by: (None)