Theorem neiss2 14614

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 14613	. . . . 5 |- ( J e. Top -> ( nei ` J ) Fn ~P X )
3		fnrel 5372	. . . . 5 |- ( ( nei ` J ) Fn ~P X -> Rel ( nei ` J ) )
4	2, 3	syl 14	. . . 4 |- ( J e. Top -> Rel ( nei ` J ) )
5		relelfvdm 5608	. . . 4 |- ( ( Rel ( nei ` J ) /\ N e. ( ( nei ` J ) ` S ) ) -> S e. dom ( nei ` J ) )
6	4, 5	sylan 283	. . 3 |- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) ) -> S e. dom ( nei ` J ) )
7		fndm 5373	. . . . . 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 2275	. . . 4 |- ( J e. Top -> ( S e. dom ( nei ` J ) <-> S e. ~P X ) )
10	9	adantr 276	. . 3 |- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) ) -> ( S e. dom ( nei ` J ) <-> S e. ~P X ) )
11	6, 10	mpbid 147	. 2 |- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) ) -> S e. ~P X )
12	11	elpwid 3627	1 |- ( ( J e. Top /\ N e. ( ( nei ` J ) ` S ) ) -> S C_ X )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1373   e. wcel 2176   C_ wss 3166   ~Pcpw 3616   U.cuni 3850   dom cdm 4675   Rel wrel 4680   Fn wfn 5266   ` cfv 5271   Topctop 14469   neicnei 14610

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-coll 4159  ax-sep 4162  ax-pow 4218  ax-pr 4253

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-top 14470  df-nei 14611

This theorem is referenced by:  neii1  14619  neii2  14621  neiss  14622  ssnei2  14629  topssnei  14634  innei  14635  neitx  14740