Theorem isptfin 21319

Description: The statement "is a point-finite cover." (Contributed by Jeff Hankins, 21-Jan-2010.)

Hypothesis

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

Assertion

Ref	Expression
isptfin	|- ( A e. B -> ( A e. PtFin <-> A. x e. X { y e. A | x e. y } e. Fin ) )

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

Allowed substitution hints:   B( x, y)   X( y)

Proof of Theorem isptfin

Dummy variable a is distinct from all other variables.

Step	Hyp	Ref	Expression
1		unieq 4444	. . . 4 |- ( a = A -> U. a = U. A )
2		isptfin.1	. . . 4 |- X = U. A
3	1, 2	syl6eqr 2674	. . 3 |- ( a = A -> U. a = X )
4		rabeq 3192	. . . 4 |- ( a = A -> { y e. a | x e. y } = { y e. A | x e. y } )
5	4	eleq1d 2686	. . 3 |- ( a = A -> ( { y e. a | x e. y } e. Fin <-> { y e. A | x e. y } e. Fin ) )
6	3, 5	raleqbidv 3152	. 2 |- ( a = A -> ( A. x e. U. a { y e. a | x e. y } e. Fin <-> A. x e. X { y e. A | x e. y } e. Fin ) )
7		df-ptfin 21309	. 2 |- PtFin = { a | A. x e. U. a { y e. a | x e. y } e. Fin }
8	6, 7	elab2g 3353	1 |- ( A e. B -> ( A e. PtFin <-> A. x e. X { y e. A | x e. y } e. Fin ) )

Syntax hints:   -> wi 4   <-> wb 196   = 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:  finptfin  21321  ptfinfin  21322  lfinpfin  21327