Theorem isnei 14096

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 14095	. . 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 2259	. 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 3194	. . . . . . 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 2491	. . . . 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 2908	. . . 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 13960	. . . . . 6 |- ( J e. Top -> X e. J )
9		elpw2g 4174	. . . . . 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 1364   e. wcel 2160   E.wrex 2469   {crab 2472   C_ wss 3144   ~Pcpw 3590   U.cuni 3824   ` cfv 5235   Topctop 13949   neicnei 14090

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-pow 4192  ax-pr 4227

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-top 13950  df-nei 14091

This theorem is referenced by:  neiint  14097  isneip  14098  neii1  14099  neii2  14101  neiss  14102  neipsm  14106  opnneissb  14107  opnssneib  14108  ssnei2  14109  innei  14115  neitx  14220