Theorem neif 14935

Description: The neighborhood function is a function from the set of the subsets of the base set of a topology. (Contributed by NM, 12-Feb-2007.) (Revised by Mario Carneiro, 11-Nov-2013.)

Hypothesis

Ref	Expression
neifval.1	|- X = U. J

Assertion

Ref	Expression
neif	|- ( J e. Top -> ( nei ` J ) Fn ~P X )

Proof of Theorem neif

Dummy variables g v x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		neifval.1	. . . . . 6 |- X = U. J
2	1	topopn 14802	. . . . 5 |- ( J e. Top -> X e. J )
3		pwexg 4276	. . . . 5 |- ( X e. J -> ~P X e. _V )
4		rabexg 4238	. . . . 5 |- ( ~P X e. _V -> { v e. ~P X | E. g e. J ( x C_ g /\ g C_ v ) } e. _V )
5	2, 3, 4	3syl 17	. . . 4 |- ( J e. Top -> { v e. ~P X | E. g e. J ( x C_ g /\ g C_ v ) } e. _V )
6	5	ralrimivw 2607	. . 3 |- ( J e. Top -> A. x e. ~P X { v e. ~P X | E. g e. J ( x C_ g /\ g C_ v ) } e. _V )
7		eqid 2231	. . . 4 |- ( x e. ~P X |-> { v e. ~P X | E. g e. J ( x C_ g /\ g C_ v ) } ) = ( x e. ~P X |-> { v e. ~P X | E. g e. J ( x C_ g /\ g C_ v ) } )
8	7	fnmpt 5466	. . 3 |- ( A. x e. ~P X { v e. ~P X | E. g e. J ( x C_ g /\ g C_ v ) } e. _V -> ( x e. ~P X |-> { v e. ~P X | E. g e. J ( x C_ g /\ g C_ v ) } ) Fn ~P X )
9	6, 8	syl 14	. 2 |- ( J e. Top -> ( x e. ~P X |-> { v e. ~P X | E. g e. J ( x C_ g /\ g C_ v ) } ) Fn ~P X )
10	1	neifval 14934	. . 3 |- ( J e. Top -> ( nei ` J ) = ( x e. ~P X |-> { v e. ~P X | E. g e. J ( x C_ g /\ g C_ v ) } ) )
11	10	fneq1d 5427	. 2 |- ( J e. Top -> ( ( nei ` J ) Fn ~P X <-> ( x e. ~P X |-> { v e. ~P X | E. g e. J ( x C_ g /\ g C_ v ) } ) Fn ~P X ) )
12	9, 11	mpbird 167	1 |- ( J e. Top -> ( nei ` J ) Fn ~P X )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2202   A.wral 2511   E.wrex 2512   {crab 2515   _Vcvv 2803   C_ wss 3201   ~Pcpw 3656   U.cuni 3898   |-> cmpt 4155   Fn wfn 5328   ` cfv 5333   Topctop 14791   neicnei 14932

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-top 14792  df-nei 14933

This theorem is referenced by:  neiss2  14936