Theorem neiss2 12154

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 12153	. . . . 5 |- ( J e. Top -> ( nei ` J ) Fn ~P X )
3		fnrel 5179	. . . . 5 |- ( ( nei ` J ) Fn ~P X -> Rel ( nei ` J ) )
4	2, 3	syl 14	. . . 4 |- ( J e. Top -> Rel ( nei ` J ) )
5		relelfvdm 5407	. . . 4 |- ( ( Rel ( nei ` J ) /\ N e. ( ( nei ` J ) ` S ) ) -> S e. dom ( nei ` J ) )
6	4, 5	sylan 279	. . 3 |- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) ) -> S e. dom ( nei ` J ) )
7		fndm 5180	. . . . . 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 2184	. . . 4 |- ( J e. Top -> ( S e. dom ( nei ` J ) <-> S e. ~P X ) )
10	9	adantr 272	. . 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 3487	1 |- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) ) -> S C_ X )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1314   e. wcel 1463   C_ wss 3037   ~Pcpw 3476   U.cuni 3702   dom cdm 4499   Rel wrel 4504   Fn wfn 5076   ` cfv 5081   Topctop 12007   neicnei 12150

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4003  ax-sep 4006  ax-pow 4058  ax-pr 4091

This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-rex 2396  df-reu 2397  df-rab 2399  df-v 2659  df-sbc 2879  df-csb 2972  df-un 3041  df-in 3043  df-ss 3050  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-iun 3781  df-br 3896  df-opab 3950  df-mpt 3951  df-id 4175  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-rn 4510  df-res 4511  df-ima 4512  df-iota 5046  df-fun 5083  df-fn 5084  df-f 5085  df-f1 5086  df-fo 5087  df-f1o 5088  df-fv 5089  df-top 12008  df-nei 12151

This theorem is referenced by:  neii1  12159  neii2  12161  neiss  12162  ssnei2  12169  topssnei  12174  innei  12175  neitx  12279