Theorem neiint 12785

Description: An intuitive definition of a neighborhood in terms of interior. (Contributed by Szymon Jaroszewicz, 18-Dec-2007.) (Revised by Mario Carneiro, 11-Nov-2013.)

Hypothesis

Ref	Expression
neifval.1	|- X = U. J

Assertion

Ref	Expression
neiint	|- ( ( J e. Top /\ S C_ X /\ N C_ X ) -> ( N e. ( ( nei ` J ) ` S ) <-> S C_ ( ( int ` J ) ` N ) ) )

Proof of Theorem neiint

Dummy variable v is distinct from all other variables.

Step	Hyp	Ref	Expression
1		neifval.1	. . . . 5 |- X = U. J
2	1	isnei 12784	. . . 4 |- ( ( J e. Top /\ S C_ X ) -> ( N e. ( ( nei ` J ) ` S ) <-> ( N C_ X /\ E. v e. J ( S C_ v /\ v C_ N ) ) ) )
3	2	3adant3 1007	. . 3 |- ( ( J e. Top /\ S C_ X /\ N C_ X ) -> ( N e. ( ( nei ` J ) ` S ) <-> ( N C_ X /\ E. v e. J ( S C_ v /\ v C_ N ) ) ) )
4	3	3anibar 1155	. 2 |- ( ( J e. Top /\ S C_ X /\ N C_ X ) -> ( N e. ( ( nei ` J ) ` S ) <-> E. v e. J ( S C_ v /\ v C_ N ) ) )
5		simprrl 529	. . . . 5 |- ( ( ( J e. Top /\ S C_ X /\ N C_ X ) /\ ( v e. J /\ ( S C_ v /\ v C_ N ) ) ) -> S C_ v )
6	1	ssntr 12762	. . . . . . 7 |- ( ( ( J e. Top /\ N C_ X ) /\ ( v e. J /\ v C_ N ) ) -> v C_ ( ( int ` J ) ` N ) )
7	6	3adantl2 1144	. . . . . 6 |- ( ( ( J e. Top /\ S C_ X /\ N C_ X ) /\ ( v e. J /\ v C_ N ) ) -> v C_ ( ( int ` J ) ` N ) )
8	7	adantrrl 478	. . . . 5 |- ( ( ( J e. Top /\ S C_ X /\ N C_ X ) /\ ( v e. J /\ ( S C_ v /\ v C_ N ) ) ) -> v C_ ( ( int ` J ) ` N ) )
9	5, 8	sstrd 3152	. . . 4 |- ( ( ( J e. Top /\ S C_ X /\ N C_ X ) /\ ( v e. J /\ ( S C_ v /\ v C_ N ) ) ) -> S C_ ( ( int ` J ) ` N ) )
10	9	rexlimdvaa 2584	. . 3 |- ( ( J e. Top /\ S C_ X /\ N C_ X ) -> ( E. v e. J ( S C_ v /\ v C_ N ) -> S C_ ( ( int ` J ) ` N ) ) )
11		simpl1 990	. . . . . 6 |- ( ( ( J e. Top /\ S C_ X /\ N C_ X ) /\ S C_ ( ( int ` J ) ` N ) ) -> J e. Top )
12		simpl3 992	. . . . . 6 |- ( ( ( J e. Top /\ S C_ X /\ N C_ X ) /\ S C_ ( ( int ` J ) ` N ) ) -> N C_ X )
13	1	ntropn 12757	. . . . . 6 |- ( ( J e. Top /\ N C_ X ) -> ( ( int ` J ) ` N ) e. J )
14	11, 12, 13	syl2anc 409	. . . . 5 |- ( ( ( J e. Top /\ S C_ X /\ N C_ X ) /\ S C_ ( ( int ` J ) ` N ) ) -> ( ( int ` J ) ` N ) e. J )
15		simpr 109	. . . . 5 |- ( ( ( J e. Top /\ S C_ X /\ N C_ X ) /\ S C_ ( ( int ` J ) ` N ) ) -> S C_ ( ( int ` J ) ` N ) )
16	1	ntrss2 12761	. . . . . 6 |- ( ( J e. Top /\ N C_ X ) -> ( ( int ` J ) ` N ) C_ N )
17	11, 12, 16	syl2anc 409	. . . . 5 |- ( ( ( J e. Top /\ S C_ X /\ N C_ X ) /\ S C_ ( ( int ` J ) ` N ) ) -> ( ( int ` J ) ` N ) C_ N )
18		sseq2 3166	. . . . . . 7 |- ( v = ( ( int ` J ) ` N ) -> ( S C_ v <-> S C_ ( ( int ` J ) ` N ) ) )
19		sseq1 3165	. . . . . . 7 |- ( v = ( ( int ` J ) ` N ) -> ( v C_ N <-> ( ( int ` J ) ` N ) C_ N ) )
20	18, 19	anbi12d 465	. . . . . 6 |- ( v = ( ( int ` J ) ` N ) -> ( ( S C_ v /\ v C_ N ) <-> ( S C_ ( ( int ` J ) ` N ) /\ ( ( int ` J ) ` N ) C_ N ) ) )
21	20	rspcev 2830	. . . . 5 |- ( ( ( ( int ` J ) ` N ) e. J /\ ( S C_ ( ( int ` J ) ` N ) /\ ( ( int ` J ) ` N ) C_ N ) ) -> E. v e. J ( S C_ v /\ v C_ N ) )
22	14, 15, 17, 21	syl12anc 1226	. . . 4 |- ( ( ( J e. Top /\ S C_ X /\ N C_ X ) /\ S C_ ( ( int ` J ) ` N ) ) -> E. v e. J ( S C_ v /\ v C_ N ) )
23	22	ex 114	. . 3 |- ( ( J e. Top /\ S C_ X /\ N C_ X ) -> ( S C_ ( ( int ` J ) ` N ) -> E. v e. J ( S C_ v /\ v C_ N ) ) )
24	10, 23	impbid 128	. 2 |- ( ( J e. Top /\ S C_ X /\ N C_ X ) -> ( E. v e. J ( S C_ v /\ v C_ N ) <-> S C_ ( ( int ` J ) ` N ) ) )
25	4, 24	bitrd 187	1 |- ( ( J e. Top /\ S C_ X /\ N C_ X ) -> ( N e. ( ( nei ` J ) ` S ) <-> S C_ ( ( int ` J ) ` N ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 968   = wceq 1343   e. wcel 2136   E.wrex 2445   C_ wss 3116   U.cuni 3789   ` cfv 5188   Topctop 12635   intcnt 12733   neicnei 12778

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-top 12636  df-ntr 12736  df-nei 12779

This theorem is referenced by:  topssnei  12802  iscnp4  12858