Theorem isneip 20909

Description: The predicate " N is a neighborhood of point P." (Contributed by NM, 26-Feb-2007.)

Hypothesis

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

Assertion

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

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

Proof of Theorem isneip

Step	Hyp	Ref	Expression
1		snssi 4339	. . 3 |- ( P e. X -> { P } C_ X )
2		neifval.1	. . . 4 |- X = U. J
3	2	isnei 20907	. . 3 |- ( ( J e. Top /\ { P } C_ X ) -> ( N e. ( ( nei ` J ) ` { P } ) <-> ( N C_ X /\ E. g e. J ( { P } C_ g /\ g C_ N ) ) ) )
4	1, 3	sylan2 491	. 2 |- ( ( J e. Top /\ P e. X ) -> ( N e. ( ( nei ` J ) ` { P } ) <-> ( N C_ X /\ E. g e. J ( { P } C_ g /\ g C_ N ) ) ) )
5		snssg 4327	. . . . . 6 |- ( P e. X -> ( P e. g <-> { P } C_ g ) )
6	5	anbi1d 741	. . . . 5 |- ( P e. X -> ( ( P e. g /\ g C_ N ) <-> ( { P } C_ g /\ g C_ N ) ) )
7	6	rexbidv 3052	. . . 4 |- ( P e. X -> ( E. g e. J ( P e. g /\ g C_ N ) <-> E. g e. J ( { P } C_ g /\ g C_ N ) ) )
8	7	anbi2d 740	. . 3 |- ( P e. X -> ( ( N C_ X /\ E. g e. J ( P e. g /\ g C_ N ) ) <-> ( N C_ X /\ E. g e. J ( { P } C_ g /\ g C_ N ) ) ) )
9	8	adantl 482	. 2 |- ( ( J e. Top /\ P e. X ) -> ( ( N C_ X /\ E. g e. J ( P e. g /\ g C_ N ) ) <-> ( N C_ X /\ E. g e. J ( { P } C_ g /\ g C_ N ) ) ) )
10	4, 9	bitr4d 271	1 |- ( ( J e. Top /\ P e. X ) -> ( N e. ( ( nei ` J ) ` { P } ) <-> ( N C_ X /\ E. g e. J ( P e. g /\ g C_ N ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   C_ wss 3574   {csn 4177   U.cuni 4436   ` cfv 5888   Topctop 20698   neicnei 20901

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-top 20699  df-nei 20902

This theorem is referenced by:  neips  20917  neindisj  20921  neindisj2  20927  neiptopnei  20936  cnpnei  21068  fbflim2  21781  cnpflf2  21804  neibl  22306  neibastop2  32356  neibastop3  32357