Theorem locfinnei 15512

Description: A point covered by a locally finite cover has a neighborhood which intersects only finitely many elements of the cover.

Hypothesis

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

Assertion

Ref	Expression
locfinnei	|- ((A e. B /\ <.J, A>. e. LocFin /\ P e. X) -> E.n e. ((nei` J)` {P}){s e. A | (s i^i n) =/= (/)} e. Fin)

Distinct variable groups:   n,s,A   B,n,s   n,J,s   P,n,s   n,X,s

Proof of Theorem locfinnei

Step	Hyp	Ref	Expression
1		locfinnei.1	. . . . . 6 |- X = U.J
2		eqid 1884	. . . . . 6 |- U.A = U.A
3	1, 2	islocfin 15506	. . . . 5 |- (A e. B -> (<.J, A>. e. LocFin <-> (J e. Top /\ X = U.A /\ A.p e. X E.n e. ((nei` J)` {p}){s e. A | (s i^i n) =/= (/)} e. Fin)))
4	3	biimpa 460	. . . 4 |- ((A e. B /\ <.J, A>. e. LocFin) -> (J e. Top /\ X = U.A /\ A.p e. X E.n e. ((nei` J)` {p}){s e. A | (s i^i n) =/= (/)} e. Fin))
5	4	simp3d 890	. . 3 |- ((A e. B /\ <.J, A>. e. LocFin) -> A.p e. X E.n e. ((nei` J)` {p}){s e. A | (s i^i n) =/= (/)} e. Fin)
6		sneq 3054	. . . . . 6 |- (p = P -> {p} = {P})
7	6	fveq2d 4685	. . . . 5 |- (p = P -> ((nei` J)` {p}) = ((nei` J)` {P}))
8	7	rexeqdv 2270	. . . 4 |- (p = P -> (E.n e. ((nei` J)` {p}){s e. A | (s i^i n) =/= (/)} e. Fin <-> E.n e. ((nei` J)` {P}){s e. A | (s i^i n) =/= (/)} e. Fin))
9	8	rcla4cv 2377	. . 3 |- (A.p e. X E.n e. ((nei` J)` {p}){s e. A | (s i^i n) =/= (/)} e. Fin -> (P e. X -> E.n e. ((nei` J)` {P}){s e. A | (s i^i n) =/= (/)} e. Fin))
10	5, 9	syl 12	. 2 |- ((A e. B /\ <.J, A>. e. LocFin) -> (P e. X -> E.n e. ((nei` J)` {P}){s e. A | (s i^i n) =/= (/)} e. Fin))
11	10	3impia 1064	1 |- ((A e. B /\ <.J, A>. e. LocFin /\ P e. X) -> E.n e. ((nei` J)` {P}){s e. A | (s i^i n) =/= (/)} e. Fin)

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300   =/= wne 2017  A.wral 2105  E.wrex 2106  {crab 2108   i^i cin 2592  (/)c0 2875  {csn 3044  <.cop 3046  U.cuni 3177  ` cfv 3998  Fincfn 5426  Topctop 8857  neicnei 8988  LocFinclocfin 15460

This theorem is referenced by:  locfincomp 15514  locfincf 15516

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-xp 4000  df-rel 4001  df-cnv 4002  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fv 4014  df-locfin 15466

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