Theorem ptfinfin 15508

Description: A point covered by a point-finite cover is only covered by finitely many elements.

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

Step	Hyp	Ref	Expression
1		ptfinfin.1	. . . . 5 |- X = U.A
2	1	isptfin 15505	. . . 4 |- (A e. PtFin -> (A e. PtFin <-> A.p e. X {x e. A | p e. x} e. Fin))
3	2	ibi 652	. . 3 |- (A e. PtFin -> A.p e. X {x e. A | p e. x} e. Fin)
4		eleq1 1957	. . . . . 6 |- (p = P -> (p e. x <-> P e. x))
5	4	rabbidv 2287	. . . . 5 |- (p = P -> {x e. A | p e. x} = {x e. A | P e. x})
6	5	eleq1d 1963	. . . 4 |- (p = P -> ({x e. A | p e. x} e. Fin <-> {x e. A | P e. x} e. Fin))
7	6	rcla4cv 2377	. . 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 12	. 2 |- (A e. PtFin -> (P e. X -> {x e. A | P e. x} e. Fin))
9	8	imp 377	1 |- ((A e. PtFin /\ P e. X) -> {x e. A | P e. x} e. Fin)

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105  {crab 2108  U.cuni 3177  Fincfn 5426  PtFincptfin 15459

This theorem is referenced by:  locfindsc 15515  comppfsc 15517

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-uni 3178  df-ptfin 15465

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