Theorem neiss2 12300

Description: A set with a neighborhood is a subset of the base set of a topology. (This theorem depends on a function's value being empty outside of its domain, but it will make later theorems simpler to state.) (Contributed by NM, 12-Feb-2007.)

Hypothesis

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

Assertion

Ref	Expression
neiss2	|- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) ) -> S C_ X )

Proof of Theorem neiss2

Step	Hyp	Ref	Expression
1		neifval.1	. . . . . 6 |- X = U. J
2	1	neif 12299	. . . . 5 |- ( J e. Top -> ( nei ` J ) Fn ~P X )
3		fnrel 5216	. . . . 5 |- ( ( nei ` J ) Fn ~P X -> Rel ( nei ` J ) )
4	2, 3	syl 14	. . . 4 |- ( J e. Top -> Rel ( nei ` J ) )
5		relelfvdm 5446	. . . 4 |- ( ( Rel ( nei ` J ) /\ N e. ( ( nei ` J ) ` S ) ) -> S e. dom ( nei ` J ) )
6	4, 5	sylan 281	. . 3 |- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) ) -> S e. dom ( nei ` J ) )
7		fndm 5217	. . . . . 6 |- ( ( nei ` J ) Fn ~P X -> dom ( nei ` J ) = ~P X )
8	2, 7	syl 14	. . . . 5 |- ( J e. Top -> dom ( nei ` J ) = ~P X )
9	8	eleq2d 2207	. . . 4 |- ( J e. Top -> ( S e. dom ( nei ` J ) <-> S e. ~P X ) )
10	9	adantr 274	. . 3 |- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) ) -> ( S e. dom ( nei ` J ) <-> S e. ~P X ) )
11	6, 10	mpbid 146	. 2 |- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) ) -> S e. ~P X )
12	11	elpwid 3516	1 |- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) ) -> S C_ X )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1331   e. wcel 1480   C_ wss 3066   ~Pcpw 3505   U.cuni 3731   dom cdm 4534   Rel wrel 4539   Fn wfn 5113   ` cfv 5118   Topctop 12153   neicnei 12296

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-pow 4093  ax-pr 4126

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-top 12154  df-nei 12297

This theorem is referenced by:  neii1  12305  neii2  12307  neiss  12308  ssnei2  12315  topssnei  12320  innei  12321  neitx  12426