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Theorem ptfinfin 22128
Description: A point covered by a point-finite cover is only covered by finitely many elements. (Contributed by Jeff Hankins, 21-Jan-2010.)
Hypothesis
Ref Expression
ptfinfin.1 𝑋 = 𝐴
Assertion
Ref Expression
ptfinfin ((𝐴 ∈ PtFin ∧ 𝑃𝑋) → {𝑥𝐴𝑃𝑥} ∈ Fin)
Distinct variable groups:   𝑥,𝐴   𝑥,𝑃   𝑥,𝑋

Proof of Theorem ptfinfin
Dummy variable 𝑝 is distinct from all other variables.
StepHypRef Expression
1 ptfinfin.1 . . . . 5 𝑋 = 𝐴
21isptfin 22125 . . . 4 (𝐴 ∈ PtFin → (𝐴 ∈ PtFin ↔ ∀𝑝𝑋 {𝑥𝐴𝑝𝑥} ∈ Fin))
32ibi 270 . . 3 (𝐴 ∈ PtFin → ∀𝑝𝑋 {𝑥𝐴𝑝𝑥} ∈ Fin)
4 eleq1 2880 . . . . . 6 (𝑝 = 𝑃 → (𝑝𝑥𝑃𝑥))
54rabbidv 3430 . . . . 5 (𝑝 = 𝑃 → {𝑥𝐴𝑝𝑥} = {𝑥𝐴𝑃𝑥})
65eleq1d 2877 . . . 4 (𝑝 = 𝑃 → ({𝑥𝐴𝑝𝑥} ∈ Fin ↔ {𝑥𝐴𝑃𝑥} ∈ Fin))
76rspccv 3571 . . 3 (∀𝑝𝑋 {𝑥𝐴𝑝𝑥} ∈ Fin → (𝑃𝑋 → {𝑥𝐴𝑃𝑥} ∈ Fin))
83, 7syl 17 . 2 (𝐴 ∈ PtFin → (𝑃𝑋 → {𝑥𝐴𝑃𝑥} ∈ Fin))
98imp 410 1 ((𝐴 ∈ PtFin ∧ 𝑃𝑋) → {𝑥𝐴𝑃𝑥} ∈ Fin)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2112  wral 3109  {crab 3113   cuni 4803  Fincfn 8496  PtFincptfin 22112
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rab 3118  df-v 3446  df-in 3891  df-ss 3901  df-uni 4804  df-ptfin 22115
This theorem is referenced by:  locfindis  22139  comppfsc  22141
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