Theorem isnei 14858

Description: The predicate "the class N is a neighborhood of S". (Contributed by FL, 25-Sep-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)

Hypothesis

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

Assertion

Ref	Expression
isnei	|- ( ( J e. Top /\ S C_ X ) -> ( N e. ( ( nei ` J ) ` S ) <-> ( N C_ X /\ E. g e. J ( S C_ g /\ g C_ N ) ) ) )

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

Proof of Theorem isnei

Dummy variable v is distinct from all other variables.

Step	Hyp	Ref	Expression
1		neifval.1	. . . 4 |- X = U. J
2	1	neival 14857	. . 3 |- ( ( J e. Top /\ S C_ X ) -> ( ( nei ` J ) ` S ) = { v e. ~P X | E. g e. J ( S C_ g /\ g C_ v ) } )
3	2	eleq2d 2299	. 2 |- ( ( J e. Top /\ S C_ X ) -> ( N e. ( ( nei ` J ) ` S ) <-> N e. { v e. ~P X | E. g e. J ( S C_ g /\ g C_ v ) } ) )
4		sseq2 3249	. . . . . . 7 |- ( v = N -> ( g C_ v <-> g C_ N ) )
5	4	anbi2d 464	. . . . . 6 |- ( v = N -> ( ( S C_ g /\ g C_ v ) <-> ( S C_ g /\ g C_ N ) ) )
6	5	rexbidv 2531	. . . . 5 |- ( v = N -> ( E. g e. J ( S C_ g /\ g C_ v ) <-> E. g e. J ( S C_ g /\ g C_ N ) ) )
7	6	elrab 2960	. . . 4 |- ( N e. { v e. ~P X | E. g e. J ( S C_ g /\ g C_ v ) } <-> ( N e. ~P X /\ E. g e. J ( S C_ g /\ g C_ N ) ) )
8	1	topopn 14722	. . . . . 6 |- ( J e. Top -> X e. J )
9		elpw2g 4244	. . . . . 6 |- ( X e. J -> ( N e. ~P X <-> N C_ X ) )
10	8, 9	syl 14	. . . . 5 |- ( J e. Top -> ( N e. ~P X <-> N C_ X ) )
11	10	anbi1d 465	. . . 4 |- ( J e. Top -> ( ( N e. ~P X /\ E. g e. J ( S C_ g /\ g C_ N ) ) <-> ( N C_ X /\ E. g e. J ( S C_ g /\ g C_ N ) ) ) )
12	7, 11	bitrid 192	. . 3 |- ( J e. Top -> ( N e. { v e. ~P X | E. g e. J ( S C_ g /\ g C_ v ) } <-> ( N C_ X /\ E. g e. J ( S C_ g /\ g C_ N ) ) ) )
13	12	adantr 276	. 2 |- ( ( J e. Top /\ S C_ X ) -> ( N e. { v e. ~P X | E. g e. J ( S C_ g /\ g C_ v ) } <-> ( N C_ X /\ E. g e. J ( S C_ g /\ g C_ N ) ) ) )
14	3, 13	bitrd 188	1 |- ( ( J e. Top /\ S C_ X ) -> ( N e. ( ( nei ` J ) ` S ) <-> ( N C_ X /\ E. g e. J ( S C_ g /\ g C_ N ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   E.wrex 2509   {crab 2512   C_ wss 3198   ~Pcpw 3650   U.cuni 3891   ` cfv 5324   Topctop 14711   neicnei 14852

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-pow 4262  ax-pr 4297

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-top 14712  df-nei 14853

This theorem is referenced by:  neiint  14859  isneip  14860  neii1  14861  neii2  14863  neiss  14864  neipsm  14868  opnneissb  14869  opnssneib  14870  ssnei2  14871  innei  14877  neitx  14982