Theorem neiss2 8992

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.)

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 4704	. . . 4 |- (N e. ((nei` J)` S) -> S e. dom (nei` J))
2	1	adantl 424	. . 3 |- ((J e. Top /\ N e. ((nei` J)` S)) -> S e. dom (nei` J))
3		neifval.1	. . . . . . 7 |- X = U.J
4	3	neif 8991	. . . . . 6 |- (J e. Top -> (nei` J) Fn ~PX)
5		fndm 4512	. . . . . 6 |- ((nei` J) Fn ~PX -> dom (nei` J) = ~PX)
6	4, 5	syl 12	. . . . 5 |- (J e. Top -> dom (nei` J) = ~PX)
7	6	eleq2d 1964	. . . 4 |- (J e. Top -> (S e. dom (nei` J) <-> S e. ~PX))
8	7	adantr 425	. . 3 |- ((J e. Top /\ N e. ((nei` J)` S)) -> (S e. dom (nei` J) <-> S e. ~PX))
9	2, 8	mpbid 212	. 2 |- ((J e. Top /\ N e. ((nei` J)` S)) -> S e. ~PX)
10		uniexg 3795	. . . . 5 |- (J e. Top -> U.J e. _V)
11	10, 3	syl5eqel 1975	. . . 4 |- (J e. Top -> X e. _V)
12		elpw2g 3463	. . . 4 |- (X e. _V -> (S e. ~PX <-> S C_ X))
13	11, 12	syl 12	. . 3 |- (J e. Top -> (S e. ~PX <-> S C_ X))
14	13	adantr 425	. 2 |- ((J e. Top /\ N e. ((nei` J)` S)) -> (S e. ~PX <-> S C_ X))
15	9, 14	mpbid 212	1 |- ((J e. Top /\ N e. ((nei` J)` S)) -> S C_ X)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  _Vcvv 2292   C_ wss 2593  ~Pcpw 3032  U.cuni 3177  dom cdm 3986   Fn wfn 3993  ` cfv 3998  Topctop 8857  neicnei 8988

This theorem is referenced by:  neii1 8997  neii2 8998  neiss 8999  ssnei2 9006  innei 9012  neiin 15418

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-fv 4014  df-nei 8989

Copyright terms: Public domain