Theorem neif 14584

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 14451	. . . . 5 |- ( J e. Top -> X e. J )
3		pwexg 4223	. . . . 5 |- ( X e. J -> ~P X e. _V )
4		rabexg 4186	. . . . 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 2579	. . 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 2204	. . . 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 5401	. . 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 14583	. . 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 5363	. 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 1372   e. wcel 2175   A.wral 2483   E.wrex 2484   {crab 2487   _Vcvv 2771   C_ wss 3165   ~Pcpw 3615   U.cuni 3849   |-> cmpt 4104   Fn wfn 5265   ` cfv 5270   Topctop 14440   neicnei 14581

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-pow 4217  ax-pr 4252

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-reu 2490  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4339  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277  df-fv 5278  df-top 14441  df-nei 14582

This theorem is referenced by:  neiss2  14585