Theorem neival 12937

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 12934	. . . 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 5498	. . 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 274	. 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 12800	. . . . 5 |- ( J e. Top -> X e. J )
6		elpw2g 4142	. . . . 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 295	. . 3 |- ( ( J e. Top /\ S C_ X ) -> S e. ~P X )
9		pwexg 4166	. . . . 5 |- ( X e. J -> ~P X e. _V )
10		rabexg 4132	. . . . 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 274	. . 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 3170	. . . . . . 7 |- ( x = S -> ( x C_ g <-> S C_ g ) )
14	13	anbi1d 462	. . . . . 6 |- ( x = S -> ( ( x C_ g /\ g C_ v ) <-> ( S C_ g /\ g C_ v ) ) )
15	14	rexbidv 2471	. . . . 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 2719	. . . 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 2170	. . . 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 5572	. . 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 409	. 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 2203	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 103   <-> wb 104   = wceq 1348   e. wcel 2141   E.wrex 2449   {crab 2452   _Vcvv 2730   C_ wss 3121   ~Pcpw 3566   U.cuni 3796   |-> cmpt 4050   ` cfv 5198   Topctop 12789   neicnei 12932

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-top 12790  df-nei 12933

This theorem is referenced by:  isnei  12938