Theorem isneip 7720

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 7718	. . 3 |- ((J e. Top /\ {P} (_ X) -> (N e. ((nei` J)` {P}) <-> (N (_ X /\ E.g e. J ({P} (_ g /\ g (_ N))))
3		snssi 2466	. . 3 |- (P e. X -> {P} (_ X)
4	2, 3	sylan2 451	. 2 |- ((J e. Top /\ P e. X) -> (N e. ((nei` J)` {P}) <-> (N (_ X /\ E.g e. J ({P} (_ g /\ g (_ N))))
5		snssg 2463	. . . . . 6 |- (P e. X -> (P e. g <-> {P} (_ g))
6	5	anbi1d 617	. . . . 5 |- (P e. X -> ((P e. g /\ g (_ N) <-> ({P} (_ g /\ g (_ N)))
7	6	rexbidv 1664	. . . 4 |- (P e. X -> (E.g e. J (P e. g /\ g (_ N) <-> E.g e. J ({P} (_ g /\ g (_ N)))
8	7	anbi2d 616	. . 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 388	. 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 531	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 956   e. wcel 958  E.wrex 1646   (_ wss 2047  {csn 2409  U.cuni 2503  ` cfv 3182  Topctop 7588  neicnei 7712

This theorem is referenced by:  neips 7727  neindisj 7731  islp2 7747  neibl 7877

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-rex 1650  df-rab 1652  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-fv 3198  df-nei 7713

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