Theorem isnei 12156

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 12155	. . 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 2184	. 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 3087	. . . . . . 7 |- ( v = N -> ( g C_ v <-> g C_ N ) )
5	4	anbi2d 457	. . . . . 6 |- ( v = N -> ( ( S C_ g /\ g C_ v ) <-> ( S C_ g /\ g C_ N ) ) )
6	5	rexbidv 2412	. . . . 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 2809	. . . 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 12018	. . . . . 6 |- ( J e. Top -> X e. J )
9		elpw2g 4041	. . . . . 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 458	. . . 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	syl5bb 191	. . 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 272	. 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 187	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 103   <-> wb 104   = wceq 1314   e. wcel 1463   E.wrex 2391   {crab 2394   C_ wss 3037   ~Pcpw 3476   U.cuni 3702   ` cfv 5081   Topctop 12007   neicnei 12150

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4003  ax-sep 4006  ax-pow 4058  ax-pr 4091

This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-rex 2396  df-reu 2397  df-rab 2399  df-v 2659  df-sbc 2879  df-csb 2972  df-un 3041  df-in 3043  df-ss 3050  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-iun 3781  df-br 3896  df-opab 3950  df-mpt 3951  df-id 4175  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-rn 4510  df-res 4511  df-ima 4512  df-iota 5046  df-fun 5083  df-fn 5084  df-f 5085  df-f1 5086  df-fo 5087  df-f1o 5088  df-fv 5089  df-top 12008  df-nei 12151

This theorem is referenced by:  neiint  12157  isneip  12158  neii1  12159  neii2  12161  neiss  12162  neipsm  12166  opnneissb  12167  opnssneib  12168  ssnei2  12169  innei  12175  neitx  12279