Theorem isnei 13729

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 13728	. . 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 2247	. 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 3181	. . . . . . 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 2478	. . . . 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 2895	. . . 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 13593	. . . . . 6 |- ( J e. Top -> X e. J )
9		elpw2g 4158	. . . . . 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 1353   e. wcel 2148   E.wrex 2456   {crab 2459   C_ wss 3131   ~Pcpw 3577   U.cuni 3811   ` cfv 5218   Topctop 13582   neicnei 13723

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-top 13583  df-nei 13724

This theorem is referenced by:  neiint  13730  isneip  13731  neii1  13732  neii2  13734  neiss  13735  neipsm  13739  opnneissb  13740  opnssneib  13741  ssnei2  13742  innei  13748  neitx  13853