Theorem neiint 14692

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 14691	. . . 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 1020	. . 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 1168	. 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 14669	. . . . . . 7 |- ( ( ( J e. Top /\ N C_ X ) /\ ( v e. J /\ v C_ N ) ) -> v C_ ( ( int ` J ) ` N ) )
7	6	3adantl2 1157	. . . . . 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 3207	. . . 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 2625	. . 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 1003	. . . . . 6 |- ( ( ( J e. Top /\ S C_ X /\ N C_ X ) /\ S C_ ( ( int ` J ) ` N ) ) -> J e. Top )
12		simpl3 1005	. . . . . 6 |- ( ( ( J e. Top /\ S C_ X /\ N C_ X ) /\ S C_ ( ( int ` J ) ` N ) ) -> N C_ X )
13	1	ntropn 14664	. . . . . 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 14668	. . . . . 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 3221	. . . . . . 7 |- ( v = ( ( int ` J ) ` N ) -> ( S C_ v <-> S C_ ( ( int ` J ) ` N ) ) )
19		sseq1 3220	. . . . . . 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 2881	. . . . 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 1248	. . . 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 981   = wceq 1373   e. wcel 2177   E.wrex 2486   C_ wss 3170   U.cuni 3856   ` cfv 5280   Topctop 14544   intcnt 14640   neicnei 14685

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4167  ax-sep 4170  ax-pow 4226  ax-pr 4261  ax-un 4488

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-iun 3935  df-br 4052  df-opab 4114  df-mpt 4115  df-id 4348  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-res 4695  df-ima 4696  df-iota 5241  df-fun 5282  df-fn 5283  df-f 5284  df-f1 5285  df-fo 5286  df-f1o 5287  df-fv 5288  df-top 14545  df-ntr 14643  df-nei 14686

This theorem is referenced by:  topssnei  14709  iscnp4  14765