Theorem isnei 14867

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 14866	. . 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 2301	. 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 3251	. . . . . . 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 2533	. . . . 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 2962	. . . 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 14731	. . . . . 6 |- ( J e. Top -> X e. J )
9		elpw2g 4246	. . . . . 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 1397   e. wcel 2202   E.wrex 2511   {crab 2514   C_ wss 3200   ~Pcpw 3652   U.cuni 3893   ` cfv 5326   Topctop 14720   neicnei 14861

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-top 14721  df-nei 14862

This theorem is referenced by:  neiint  14868  isneip  14869  neii1  14870  neii2  14872  neiss  14873  neipsm  14877  opnneissb  14878  opnssneib  14879  ssnei2  14880  innei  14886  neitx  14991