Theorem innei 14399

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 2196	. . . . 5 |- U. J = U. J
2	1	neii1 14383	. . . 4 |- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) ) -> N C_ U. J )
3		ssinss1 3392	. . . 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 14385	. . . . 5 |- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) ) -> E. h e. J ( S C_ h /\ h C_ N ) )
7		neii2 14385	. . . . 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 14239	. . . . . . . . . . 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 3385	. . . . . . . . . . . . 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 3391	. . . . . . . . . . . 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 3207	. . . . . . . . . . . 12 |- ( g = ( h i^i v ) -> ( S C_ g <-> S C_ ( h i^i v ) ) )
17		sseq1 3206	. . . . . . . . . . . 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 2868	. . . . . . . . . 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 2614	. . . . . 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 2617	. . . . 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 14378	. . . 4 |- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) ) -> S C_ U. J )
29	1	isnei 14380	. . . 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 2167   E.wrex 2476   i^i cin 3156   C_ wss 3157   U.cuni 3839   ` cfv 5258   Topctop 14233   neicnei 14374

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-top 14234  df-nei 14375

This theorem is referenced by: (None)