Theorem innei 20929

Description: The intersection of two neighborhoods of a set is also a neighborhood of the set. Proposition Vii of [BourbakiTop1] p. I.3 . (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 2622	. . . . 5 |- U. J = U. J
2	1	neii1 20910	. . . 4 |- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) ) -> N C_ U. J )
3		ssinss1 3841	. . . 4 |- ( N C_ U. J -> ( N i^i M ) C_ U. J )
4	2, 3	syl 17	. . 3 |- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) ) -> ( N i^i M ) C_ U. J )
5	4	3adant3 1081	. 2 |- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) /\ M e. ( ( nei ` J ) ` S ) ) -> ( N i^i M ) C_ U. J )
6		neii2 20912	. . . . 5 |- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) ) -> E. h e. J ( S C_ h /\ h C_ N ) )
7		neii2 20912	. . . . 5 |- ( ( J e. Top /\ M e. ( ( nei ` J ) ` S ) ) -> E. v e. J ( S C_ v /\ v C_ M ) )
8	6, 7	anim12dan 882	. . . 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 20704	. . . . . . . . . . . 12 |- ( ( J e. Top /\ h e. J /\ v e. J ) -> ( h i^i v ) e. J )
10	9	3expa 1265	. . . . . . . . . . 11 |- ( ( ( J e. Top /\ h e. J ) /\ v e. J ) -> ( h i^i v ) e. J )
11		ssin 3835	. . . . . . . . . . . . . 14 |- ( ( S C_ h /\ S C_ v ) <-> S C_ ( h i^i v ) )
12	11	biimpi 206	. . . . . . . . . . . . 13 |- ( ( S C_ h /\ S C_ v ) -> S C_ ( h i^i v ) )
13		ss2in 3840	. . . . . . . . . . . . 13 |- ( ( h C_ N /\ v C_ M ) -> ( h i^i v ) C_ ( N i^i M ) )
14	12, 13	anim12i 590	. . . . . . . . . . . 12 |- ( ( ( 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 869	. . . . . . . . . . 11 |- ( ( ( 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 3627	. . . . . . . . . . . . 13 |- ( g = ( h i^i v ) -> ( S C_ g <-> S C_ ( h i^i v ) ) )
17		sseq1 3626	. . . . . . . . . . . . 13 |- ( g = ( h i^i v ) -> ( g C_ ( N i^i M ) <-> ( h i^i v ) C_ ( N i^i M ) ) )
18	16, 17	anbi12d 747	. . . . . . . . . . . 12 |- ( 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 3309	. . . . . . . . . . 11 |- ( ( ( 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 494	. . . . . . . . . 10 |- ( ( ( ( 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 643	. . . . . . . . 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 ) ) ) )
22	21	an32s 846	. . . . . . . 8 |- ( ( ( ( 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 3031	. . . . . . 7 |- ( ( ( 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	ex 450	. . . . . 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 ) ) ) ) )
25	24	rexlimdva 3031	. . . . 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 ) ) ) ) )
26	25	imp32 449	. . . 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 ) ) )
27	8, 26	syldan 487	. . 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 ) ) )
28	27	3impb 1260	. 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 ) ) )
29	1	neiss2 20905	. . . 4 |- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) ) -> S C_ U. J )
30	1	isnei 20907	. . . 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 ) ) ) ) )
31	29, 30	syldan 487	. . 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 ) ) ) ) )
32	31	3adant3 1081	. 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 ) ) ) ) )
33	5, 28, 32	mpbir2and 957	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   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   E.wrex 2913   i^i cin 3573   C_ wss 3574   U.cuni 4436   ` cfv 5888   Topctop 20698   neicnei 20901

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-top 20699  df-nei 20902

This theorem is referenced by:  neifil  21684  neificl  33549