Theorem neival 14730

Description: Value of the set of neighborhoods of a subset of the base set of a topology. (Contributed by NM, 11-Feb-2007.) (Revised by Mario Carneiro, 11-Nov-2013.)

Hypothesis

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

Assertion

Ref	Expression
neival	|- ( ( J e. Top /\ S C_ X ) -> ( ( nei ` J ) ` S ) = { v e. ~P X | E. g e. J ( S C_ g /\ g C_ v ) } )

Distinct variable groups:   v, g, J   S, g, v   g, X, v

Proof of Theorem neival

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		neifval.1	. . . . 5 |- X = U. J
2	1	neifval 14727	. . . 4 |- ( J e. Top -> ( nei ` J ) = ( x e. ~P X |-> { v e. ~P X | E. g e. J ( x C_ g /\ g C_ v ) } ) )
3	2	fveq1d 5601	. . 3 |- ( J e. Top -> ( ( nei ` J ) ` S ) = ( ( x e. ~P X |-> { v e. ~P X | E. g e. J ( x C_ g /\ g C_ v ) } ) ` S ) )
4	3	adantr 276	. 2 |- ( ( J e. Top /\ S C_ X ) -> ( ( nei ` J ) ` S ) = ( ( x e. ~P X |-> { v e. ~P X | E. g e. J ( x C_ g /\ g C_ v ) } ) ` S ) )
5	1	topopn 14595	. . . . 5 |- ( J e. Top -> X e. J )
6		elpw2g 4216	. . . . 5 |- ( X e. J -> ( S e. ~P X <-> S C_ X ) )
7	5, 6	syl 14	. . . 4 |- ( J e. Top -> ( S e. ~P X <-> S C_ X ) )
8	7	biimpar 297	. . 3 |- ( ( J e. Top /\ S C_ X ) -> S e. ~P X )
9		pwexg 4240	. . . . 5 |- ( X e. J -> ~P X e. _V )
10		rabexg 4203	. . . . 5 |- ( ~P X e. _V -> { v e. ~P X | E. g e. J ( S C_ g /\ g C_ v ) } e. _V )
11	5, 9, 10	3syl 17	. . . 4 |- ( J e. Top -> { v e. ~P X | E. g e. J ( S C_ g /\ g C_ v ) } e. _V )
12	11	adantr 276	. . 3 |- ( ( J e. Top /\ S C_ X ) -> { v e. ~P X | E. g e. J ( S C_ g /\ g C_ v ) } e. _V )
13		sseq1 3224	. . . . . . 7 |- ( x = S -> ( x C_ g <-> S C_ g ) )
14	13	anbi1d 465	. . . . . 6 |- ( x = S -> ( ( x C_ g /\ g C_ v ) <-> ( S C_ g /\ g C_ v ) ) )
15	14	rexbidv 2509	. . . . 5 |- ( x = S -> ( E. g e. J ( x C_ g /\ g C_ v ) <-> E. g e. J ( S C_ g /\ g C_ v ) ) )
16	15	rabbidv 2765	. . . 4 |- ( x = S -> { v e. ~P X | E. g e. J ( x C_ g /\ g C_ v ) } = { v e. ~P X | E. g e. J ( S C_ g /\ g C_ v ) } )
17		eqid 2207	. . . 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 ) } )
18	16, 17	fvmptg 5678	. . 3 |- ( ( S e. ~P X /\ { v e. ~P X | E. g e. J ( S 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 ) } ) ` S ) = { v e. ~P X | E. g e. J ( S C_ g /\ g C_ v ) } )
19	8, 12, 18	syl2anc 411	. 2 |- ( ( J e. Top /\ S C_ X ) -> ( ( x e. ~P X |-> { v e. ~P X | E. g e. J ( x C_ g /\ g C_ v ) } ) ` S ) = { v e. ~P X | E. g e. J ( S C_ g /\ g C_ v ) } )
20	4, 19	eqtrd 2240	1 |- ( ( J e. Top /\ S C_ X ) -> ( ( nei ` J ) ` S ) = { v e. ~P X | E. g e. J ( S C_ g /\ g C_ v ) } )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1373   e. wcel 2178   E.wrex 2487   {crab 2490   _Vcvv 2776   C_ wss 3174   ~Pcpw 3626   U.cuni 3864   |-> cmpt 4121   ` cfv 5290   Topctop 14584   neicnei 14725

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-pow 4234  ax-pr 4269

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-top 14585  df-nei 14726

This theorem is referenced by:  isnei  14731