Theorem isnei 7715

Description: The predicate "N is a neighborhood of S." (Contributed by FL, 25-Sep-2006.)

Hypothesis

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

Assertion

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

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

Proof of Theorem isnei

Step	Hyp	Ref	Expression
1		neifval.1	. . . 4 |- X = U.J
2	1	neival 7714	. . 3 |- ((J e. Top /\ S (_ X) -> ((nei` J)` S) = {v | (v (_ X /\ E.g e. J (S (_ g /\ g (_ v))})
3	2	eleq2d 1544	. 2 |- ((J e. Top /\ S (_ X) -> (N e. ((nei` J)` S) <-> N e. {v | (v (_ X /\ E.g e. J (S (_ g /\ g (_ v))}))
4		ssexg 2726	. . . . . . 7 |- ((N (_ X /\ X e. V) -> N e. V)
5		uniexg 2877	. . . . . . . 8 |- (J e. Top -> U.J e. V)
6	5, 1	syl5eqel 1555	. . . . . . 7 |- (J e. Top -> X e. V)
7	4, 6	sylan2 453	. . . . . 6 |- ((N (_ X /\ J e. Top) -> N e. V)
8	7	expcom 374	. . . . 5 |- (J e. Top -> (N (_ X -> N e. V))
9	8	adantr 391	. . . 4 |- ((J e. Top /\ S (_ X) -> (N (_ X -> N e. V))
10	9	adantrd 393	. . 3 |- ((J e. Top /\ S (_ X) -> ((N (_ X /\ E.g e. J (S (_ g /\ g (_ N)) -> N e. V))
11		sseq1 2085	. . . . 5 |- (v = N -> (v (_ X <-> N (_ X))
12		sseq2 2086	. . . . . . 7 |- (v = N -> (g (_ v <-> g (_ N))
13	12	anbi2d 618	. . . . . 6 |- (v = N -> ((S (_ g /\ g (_ v) <-> (S (_ g /\ g (_ N)))
14	13	rexbidv 1667	. . . . 5 |- (v = N -> (E.g e. J (S (_ g /\ g (_ v) <-> E.g e. J (S (_ g /\ g (_ N)))
15	11, 14	anbi12d 630	. . . 4 |- (v = N -> ((v (_ X /\ E.g e. J (S (_ g /\ g (_ v)) <-> (N (_ X /\ E.g e. J (S (_ g /\ g (_ N))))
16	15	elab3g 1905	. . 3 |- (((N (_ X /\ E.g e. J (S (_ g /\ g (_ N)) -> N e. V) -> (N e. {v | (v (_ X /\ E.g e. J (S (_ g /\ g (_ v))} <-> (N (_ X /\ E.g e. J (S (_ g /\ g (_ N))))
17	10, 16	syl 10	. 2 |- ((J e. Top /\ S (_ X) -> (N e. {v | (v (_ X /\ E.g e. J (S (_ g /\ g (_ v))} <-> (N (_ X /\ E.g e. J (S (_ g /\ g (_ N))))
18	3, 17	bitrd 530	1 |- ((J e. Top /\ S (_ X) -> (N e. ((nei` J)` S) <-> (N (_ X /\ E.g e. J (S (_ g /\ g (_ N))))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 958   e. wcel 960  {cab 1466  E.wrex 1649  Vcvv 1814   (_ wss 2050  U.cuni 2507  ` cfv 3188  Topctop 7590  neicnei 7709

This theorem is referenced by:  neiint 7716  isneip 7717  neii1 7718  neii2 7719  neiss 7720  neips 7724  opnneissb 7725  opnssneib 7726  ssnei2 7727  innei 7733

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-rex 1653  df-rab 1655  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-fv 3204  df-nei 7710

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