Theorem neif 13726

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 13593	. . . . 5 |- ( J e. Top -> X e. J )
3		pwexg 4182	. . . . 5 |- ( X e. J -> ~P X e. _V )
4		rabexg 4148	. . . . 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 2551	. . 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 2177	. . . 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 5344	. . 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 13725	. . 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 5308	. 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 1353   e. wcel 2148   A.wral 2455   E.wrex 2456   {crab 2459   _Vcvv 2739   C_ wss 3131   ~Pcpw 3577   U.cuni 3811   |-> cmpt 4066   Fn wfn 5213   ` cfv 5218   Topctop 13582   neicnei 13723

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-top 13583  df-nei 13724

This theorem is referenced by:  neiss2  13727