Theorem isptfin 29754

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 4246	. . . 4 |- ( a = A -> U. a = U. A )
2		isptfin.1	. . . 4 |- X = U. A
3	1, 2	syl6eqr 2519	. . 3 |- ( a = A -> U. a = X )
4		rabeq 3100	. . . 4 |- ( a = A -> { y e. a | x e. y } = { y e. A | x e. y } )
5	4	eleq1d 2529	. . 3 |- ( a = A -> ( { y e. a | x e. y } e. Fin <-> { y e. A | x e. y } e. Fin ) )
6	3, 5	raleqbidv 3065	. 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 29724	. 2 |- PtFin = { a | A. x e. U. a { y e. a | x e. y } e. Fin }
8	6, 7	elab2g 3245	1 |- ( A e. B -> ( A e. PtFin <-> A. x e. X { y e. A | x e. y } e. Fin ) )

Syntax hints:   -> wi 4   <-> wb 184   = wceq 1374   e. wcel 1762   A.wral 2807   {crab 2811   U.cuni 4238   Fincfn 7506   PtFincptfin 29720

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-uni 4239  df-ptfin 29724

This theorem is referenced by:  finptfin  29756  ptfinfin  29757  lfinpfin  29762