Theorem neiss2 12683

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 12682	. . . . 5 |- ( J e. Top -> ( nei ` J ) Fn ~P X )
3		fnrel 5280	. . . . 5 |- ( ( nei ` J ) Fn ~P X -> Rel ( nei ` J ) )
4	2, 3	syl 14	. . . 4 |- ( J e. Top -> Rel ( nei ` J ) )
5		relelfvdm 5512	. . . 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 5281	. . . . . 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 2234	. . . 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 3564	1 |- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) ) -> S C_ X )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1342   e. wcel 2135   C_ wss 3111   ~Pcpw 3553   U.cuni 3783   dom cdm 4598   Rel wrel 4603   Fn wfn 5177   ` cfv 5182   Topctop 12536   neicnei 12679

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 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-14 2138  ax-ext 2146  ax-coll 4091  ax-sep 4094  ax-pow 4147  ax-pr 4181

This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ral 2447  df-rex 2448  df-reu 2449  df-rab 2451  df-v 2723  df-sbc 2947  df-csb 3041  df-un 3115  df-in 3117  df-ss 3124  df-pw 3555  df-sn 3576  df-pr 3577  df-op 3579  df-uni 3784  df-iun 3862  df-br 3977  df-opab 4038  df-mpt 4039  df-id 4265  df-xp 4604  df-rel 4605  df-cnv 4606  df-co 4607  df-dm 4608  df-rn 4609  df-res 4610  df-ima 4611  df-iota 5147  df-fun 5184  df-fn 5185  df-f 5186  df-f1 5187  df-fo 5188  df-f1o 5189  df-fv 5190  df-top 12537  df-nei 12680

This theorem is referenced by:  neii1  12688  neii2  12690  neiss  12691  ssnei2  12698  topssnei  12703  innei  12704  neitx  12809