Theorem neiint 14381

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 14380	. . . 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 1019	. . 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 1167	. 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 539	. . . . 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 14358	. . . . . . 7 |- ( ( ( J e. Top /\ N C_ X ) /\ ( v e. J /\ v C_ N ) ) -> v C_ ( ( int ` J ) ` N ) )
7	6	3adantl2 1156	. . . . . 6 |- ( ( ( J e. Top /\ S C_ X /\ N C_ X ) /\ ( v e. J /\ v C_ N ) ) -> v C_ ( ( int ` J ) ` N ) )
8	7	adantrrl 486	. . . . 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 3193	. . . 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 2615	. . 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 1002	. . . . . 6 |- ( ( ( J e. Top /\ S C_ X /\ N C_ X ) /\ S C_ ( ( int ` J ) ` N ) ) -> J e. Top )
12		simpl3 1004	. . . . . 6 |- ( ( ( J e. Top /\ S C_ X /\ N C_ X ) /\ S C_ ( ( int ` J ) ` N ) ) -> N C_ X )
13	1	ntropn 14353	. . . . . 6 |- ( ( J e. Top /\ N C_ X ) -> ( ( int ` J ) ` N ) e. J )
14	11, 12, 13	syl2anc 411	. . . . 5 |- ( ( ( J e. Top /\ S C_ X /\ N C_ X ) /\ S C_ ( ( int ` J ) ` N ) ) -> ( ( int ` J ) ` N ) e. J )
15		simpr 110	. . . . 5 |- ( ( ( J e. Top /\ S C_ X /\ N C_ X ) /\ S C_ ( ( int ` J ) ` N ) ) -> S C_ ( ( int ` J ) ` N ) )
16	1	ntrss2 14357	. . . . . 6 |- ( ( J e. Top /\ N C_ X ) -> ( ( int ` J ) ` N ) C_ N )
17	11, 12, 16	syl2anc 411	. . . . 5 |- ( ( ( J e. Top /\ S C_ X /\ N C_ X ) /\ S C_ ( ( int ` J ) ` N ) ) -> ( ( int ` J ) ` N ) C_ N )
18		sseq2 3207	. . . . . . 7 |- ( v = ( ( int ` J ) ` N ) -> ( S C_ v <-> S C_ ( ( int ` J ) ` N ) ) )
19		sseq1 3206	. . . . . . 7 |- ( v = ( ( int ` J ) ` N ) -> ( v C_ N <-> ( ( int ` J ) ` N ) C_ N ) )
20	18, 19	anbi12d 473	. . . . . 6 |- ( v = ( ( int ` J ) ` N ) -> ( ( S C_ v /\ v C_ N ) <-> ( S C_ ( ( int ` J ) ` N ) /\ ( ( int ` J ) ` N ) C_ N ) ) )
21	20	rspcev 2868	. . . . 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 1247	. . . 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 115	. . 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 129	. 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 188	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 104   <-> wb 105   /\ w3a 980   = wceq 1364   e. wcel 2167   E.wrex 2476   C_ wss 3157   U.cuni 3839   ` cfv 5258   Topctop 14233   intcnt 14329   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-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468

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-ntr 14332  df-nei 14375

This theorem is referenced by:  topssnei  14398  iscnp4  14454