Theorem neiss2 20905

Description: A set with a neighborhood is a subset of the topology's base set. (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		elfvdm 6220	. . . 4 |- ( N e. ( ( nei ` J ) ` S ) -> S e. dom ( nei ` J ) )
2	1	adantl 482	. . 3 |- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) ) -> S e. dom ( nei ` J ) )
3		neifval.1	. . . . . . 7 |- X = U. J
4	3	neif 20904	. . . . . 6 |- ( J e. Top -> ( nei ` J ) Fn ~P X )
5		fndm 5990	. . . . . 6 |- ( ( nei ` J ) Fn ~P X -> dom ( nei ` J ) = ~P X )
6	4, 5	syl 17	. . . . 5 |- ( J e. Top -> dom ( nei ` J ) = ~P X )
7	6	eleq2d 2687	. . . 4 |- ( J e. Top -> ( S e. dom ( nei ` J ) <-> S e. ~P X ) )
8	7	adantr 481	. . 3 |- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) ) -> ( S e. dom ( nei ` J ) <-> S e. ~P X ) )
9	2, 8	mpbid 222	. 2 |- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) ) -> S e. ~P X )
10	9	elpwid 4170	1 |- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) ) -> S C_ X )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   C_ wss 3574   ~Pcpw 4158   U.cuni 4436   dom cdm 5114   Fn wfn 5883   ` cfv 5888   Topctop 20698   neicnei 20901

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-top 20699  df-nei 20902

This theorem is referenced by:  neii1  20910  neii2  20912  neiss  20913  ssnei2  20920  topssnei  20928  innei  20929  neitx  21410  cvmlift2lem12  31296  neiin  32327