Theorem isptfin 28593

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 4120	. . . 4 |- ( a = A -> U. a = U. A )
2		isptfin.1	. . . 4 |- X = U. A
3	1, 2	syl6eqr 2493	. . 3 |- ( a = A -> U. a = X )
4		rabeq 2987	. . . 4 |- ( a = A -> { y e. a | x e. y } = { y e. A | x e. y } )
5	4	eleq1d 2509	. . 3 |- ( a = A -> ( { y e. a | x e. y } e. Fin <-> { y e. A | x e. y } e. Fin ) )
6	3, 5	raleqbidv 2952	. 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 28563	. 2 |- PtFin = { a | A. x e. U. a { y e. a | x e. y } e. Fin }
8	6, 7	elab2g 3129	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 1369   e. wcel 1756   A.wral 2736   {crab 2740   U.cuni 4112   Fincfn 7331   PtFincptfin 28559

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ral 2741  df-rex 2742  df-rab 2745  df-v 2995  df-uni 4113  df-ptfin 28563

This theorem is referenced by:  finptfin  28595  ptfinfin  28596  lfinpfin  28601