Theorem neiint 14535

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 14534	. . . 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 14512	. . . . . . 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 3202	. . . 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 2623	. . 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 14507	. . . . . 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 14511	. . . . . 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 3216	. . . . . . 7 |- ( v = ( ( int ` J ) ` N ) -> ( S C_ v <-> S C_ ( ( int ` J ) ` N ) ) )
19		sseq1 3215	. . . . . . 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 2876	. . . . 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 1372   e. wcel 2175   E.wrex 2484   C_ wss 3165   U.cuni 3849   ` cfv 5268   Topctop 14387   intcnt 14483   neicnei 14528

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4478

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-reu 2490  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4338  df-xp 4679  df-rel 4680  df-cnv 4681  df-co 4682  df-dm 4683  df-rn 4684  df-res 4685  df-ima 4686  df-iota 5229  df-fun 5270  df-fn 5271  df-f 5272  df-f1 5273  df-fo 5274  df-f1o 5275  df-fv 5276  df-top 14388  df-ntr 14486  df-nei 14529

This theorem is referenced by:  topssnei  14552  iscnp4  14608