Theorem ptfinfin 21322

Description: A point covered by a point-finite cover is only covered by finitely many elements. (Contributed by Jeff Hankins, 21-Jan-2010.)

Hypothesis

Ref	Expression
ptfinfin.1	|- X = U. A

Assertion

Ref	Expression
ptfinfin	|- ( ( A e. PtFin /\ P e. X ) -> { x e. A | P e. x } e. Fin )

Distinct variable groups:   x, A   x, P   x, X

Proof of Theorem ptfinfin

Dummy variable p is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ptfinfin.1	. . . . 5 |- X = U. A
2	1	isptfin 21319	. . . 4 |- ( A e. PtFin -> ( A e. PtFin <-> A. p e. X { x e. A | p e. x } e. Fin ) )
3	2	ibi 256	. . 3 |- ( A e. PtFin -> A. p e. X { x e. A | p e. x } e. Fin )
4		eleq1 2689	. . . . . 6 |- ( p = P -> ( p e. x <-> P e. x ) )
5	4	rabbidv 3189	. . . . 5 |- ( p = P -> { x e. A | p e. x } = { x e. A | P e. x } )
6	5	eleq1d 2686	. . . 4 |- ( p = P -> ( { x e. A | p e. x } e. Fin <-> { x e. A | P e. x } e. Fin ) )
7	6	rspccv 3306	. . 3 |- ( A. p e. X { x e. A | p e. x } e. Fin -> ( P e. X -> { x e. A | P e. x } e. Fin ) )
8	3, 7	syl 17	. 2 |- ( A e. PtFin -> ( P e. X -> { x e. A | P e. x } e. Fin ) )
9	8	imp 445	1 |- ( ( A e. PtFin /\ P e. X ) -> { x e. A | P e. x } e. Fin )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   U.cuni 4436   Fincfn 7955   PtFincptfin 21306

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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-uni 4437  df-ptfin 21309

This theorem is referenced by:  locfindis  21333  comppfsc  21335