Theorem innei 14331

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 2193	. . . . 5 |- U. J = U. J
2	1	neii1 14315	. . . 4 |- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) ) -> N C_ U. J )
3		ssinss1 3388	. . . 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 1019	. 2 |- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) /\ M e. ( ( nei ` J ) ` S ) ) -> ( N i^i M ) C_ U. J )
6		neii2 14317	. . . . 5 |- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) ) -> E. h e. J ( S C_ h /\ h C_ N ) )
7		neii2 14317	. . . . 5 |- ( ( J e. Top /\ M e. ( ( nei ` J ) ` S ) ) -> E. v e. J ( S C_ v /\ v C_ M ) )
8	6, 7	anim12dan 600	. . . 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 14171	. . . . . . . . . . 11 |- ( ( J e. Top /\ h e. J /\ v e. J ) -> ( h i^i v ) e. J )
10	9	3expa 1205	. . . . . . . . . 10 |- ( ( ( J e. Top /\ h e. J ) /\ v e. J ) -> ( h i^i v ) e. J )
11		ssin 3381	. . . . . . . . . . . . 13 |- ( ( S C_ h /\ S C_ v ) <-> S C_ ( h i^i v ) )
12	11	biimpi 120	. . . . . . . . . . . 12 |- ( ( S C_ h /\ S C_ v ) -> S C_ ( h i^i v ) )
13		ss2in 3387	. . . . . . . . . . . 12 |- ( ( h C_ N /\ v C_ M ) -> ( h i^i v ) C_ ( N i^i M ) )
14	12, 13	anim12i 338	. . . . . . . . . . 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 588	. . . . . . . . . 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 3203	. . . . . . . . . . . 12 |- ( g = ( h i^i v ) -> ( S C_ g <-> S C_ ( h i^i v ) ) )
17		sseq1 3202	. . . . . . . . . . . 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 473	. . . . . . . . . . 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 2864	. . . . . . . . . 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 289	. . . . . . . . 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 375	. . . . . . . 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 568	. . . . . . 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 2611	. . . . . 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 2614	. . . . 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 257	. . . 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 282	. . 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 1201	. 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 14310	. . . 4 |- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) ) -> S C_ U. J )
29	1	isnei 14312	. . . 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 282	. . 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 1019	. 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 946	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 104   <-> wb 105   /\ w3a 980   = wceq 1364   e. wcel 2164   E.wrex 2473   i^i cin 3152   C_ wss 3153   U.cuni 3835   ` cfv 5254   Topctop 14165   neicnei 14306

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-top 14166  df-nei 14307

This theorem is referenced by: (None)