Theorem neiint 15169

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 15168	. . . 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 1048	. . 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 1196	. 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 545	. . . . 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 15146	. . . . . . 7 |- ( ( ( J e. Top /\ N C_ X ) /\ ( v e. J /\ v C_ N ) ) -> v C_ ( ( int ` J ) ` N ) )
7	6	3adantl2 1185	. . . . . 6 |- ( ( ( J e. Top /\ S C_ X /\ N C_ X ) /\ ( v e. J /\ v C_ N ) ) -> v C_ ( ( int ` J ) ` N ) )
8	7	adantrrl 490	. . . . 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 3258	. . . 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 2669	. . 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 1031	. . . . . 6 |- ( ( ( J e. Top /\ S C_ X /\ N C_ X ) /\ S C_ ( ( int ` J ) ` N ) ) -> J e. Top )
12		simpl3 1033	. . . . . 6 |- ( ( ( J e. Top /\ S C_ X /\ N C_ X ) /\ S C_ ( ( int ` J ) ` N ) ) -> N C_ X )
13	1	ntropn 15141	. . . . . 6 |- ( ( J e. Top /\ N C_ X ) -> ( ( int ` J ) ` N ) e. J )
14	11, 12, 13	syl2anc 415	. . . . 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 15145	. . . . . 6 |- ( ( J e. Top /\ N C_ X ) -> ( ( int ` J ) ` N ) C_ N )
17	11, 12, 16	syl2anc 415	. . . . 5 |- ( ( ( J e. Top /\ S C_ X /\ N C_ X ) /\ S C_ ( ( int ` J ) ` N ) ) -> ( ( int ` J ) ` N ) C_ N )
18		sseq2 3272	. . . . . . 7 |- ( v = ( ( int ` J ) ` N ) -> ( S C_ v <-> S C_ ( ( int ` J ) ` N ) ) )
19		sseq1 3271	. . . . . . 7 |- ( v = ( ( int ` J ) ` N ) -> ( v C_ N <-> ( ( int ` J ) ` N ) C_ N ) )
20	18, 19	anbi12d 477	. . . . . 6 |- ( v = ( ( int ` J ) ` N ) -> ( ( S C_ v /\ v C_ N ) <-> ( S C_ ( ( int ` J ) ` N ) /\ ( ( int ` J ) ` N ) C_ N ) ) )
21	20	rspcev 2929	. . . . 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 1276	. . . 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 1009   = wceq 1402   e. wcel 2209   E.wrex 2529   C_ wss 3220   U.cuni 3930   ` cfv 5372   Topctop 15021   intcnt 15117   neicnei 15162

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-top 15022  df-ntr 15120  df-nei 15163

This theorem is referenced by:  topssnei  15186  iscnp4  15242