Theorem isneip 7717

Description: The predicate "N is a neighborhood of point P."

Hypothesis

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

Assertion

Ref	Expression
isneip	|- ((J e. Top /\ P e. X) -> (N e. ((nei` J)` {P}) <-> (N (_ X /\ E.g e. J (P e. g /\ g (_ N))))

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

Proof of Theorem isneip

Step	Hyp	Ref	Expression
1		neifval.1	. . . 4 |- X = U.J
2	1	isnei 7715	. . 3 |- ((J e. Top /\ {P} (_ X) -> (N e. ((nei` J)` {P}) <-> (N (_ X /\ E.g e. J ({P} (_ g /\ g (_ N))))
3		snssi 2470	. . 3 |- (P e. X -> {P} (_ X)
4	2, 3	sylan2 453	. 2 |- ((J e. Top /\ P e. X) -> (N e. ((nei` J)` {P}) <-> (N (_ X /\ E.g e. J ({P} (_ g /\ g (_ N))))
5		snssg 2467	. . . . . 6 |- (P e. X -> (P e. g <-> {P} (_ g))
6	5	anbi1d 619	. . . . 5 |- (P e. X -> ((P e. g /\ g (_ N) <-> ({P} (_ g /\ g (_ N)))
7	6	rexbidv 1667	. . . 4 |- (P e. X -> (E.g e. J (P e. g /\ g (_ N) <-> E.g e. J ({P} (_ g /\ g (_ N)))
8	7	anbi2d 618	. . 3 |- (P e. X -> ((N (_ X /\ E.g e. J (P e. g /\ g (_ N)) <-> (N (_ X /\ E.g e. J ({P} (_ g /\ g (_ N))))
9	8	adantl 390	. 2 |- ((J e. Top /\ P e. X) -> ((N (_ X /\ E.g e. J (P e. g /\ g (_ N)) <-> (N (_ X /\ E.g e. J ({P} (_ g /\ g (_ N))))
10	4, 9	bitr4d 533	1 |- ((J e. Top /\ P e. X) -> (N e. ((nei` J)` {P}) <-> (N (_ X /\ E.g e. J (P e. g /\ g (_ N))))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 958   e. wcel 960  E.wrex 1649   (_ wss 2050  {csn 2413  U.cuni 2507  ` cfv 3188  Topctop 7590  neicnei 7709

This theorem is referenced by:  neips 7724  neindisj 7728  islp2 7744  neibl 7874

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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