Theorem isptfin 20143

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 4259	. . . 4 |- ( a = A -> U. a = U. A )
2		isptfin.1	. . . 4 |- X = U. A
3	1, 2	syl6eqr 2516	. . 3 |- ( a = A -> U. a = X )
4		rabeq 3103	. . . 4 |- ( a = A -> { y e. a | x e. y } = { y e. A | x e. y } )
5	4	eleq1d 2526	. . 3 |- ( a = A -> ( { y e. a | x e. y } e. Fin <-> { y e. A | x e. y } e. Fin ) )
6	3, 5	raleqbidv 3068	. 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 20133	. 2 |- PtFin = { a | A. x e. U. a { y e. a | x e. y } e. Fin }
8	6, 7	elab2g 3248	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 1395   e. wcel 1819   A.wral 2807   {crab 2811   U.cuni 4251   Fincfn 7535   PtFincptfin 20130

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-uni 4252  df-ptfin 20133

This theorem is referenced by:  finptfin  20145  ptfinfin  20146  lfinpfin  20151