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Theorem ptfinfin 22798
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 22795 . . . 4 (𝐴 ∈ PtFin → (𝐴 ∈ PtFin ↔ ∀𝑝𝑋 {𝑥𝐴𝑝𝑥} ∈ Fin))
32ibi 267 . . 3 (𝐴 ∈ PtFin → ∀𝑝𝑋 {𝑥𝐴𝑝𝑥} ∈ Fin)
4 eleq1 2826 . . . . . 6 (𝑝 = 𝑃 → (𝑝𝑥𝑃𝑥))
54rabbidv 3414 . . . . 5 (𝑝 = 𝑃 → {𝑥𝐴𝑝𝑥} = {𝑥𝐴𝑃𝑥})
65eleq1d 2823 . . . 4 (𝑝 = 𝑃 → ({𝑥𝐴𝑝𝑥} ∈ Fin ↔ {𝑥𝐴𝑃𝑥} ∈ Fin))
76rspccv 3577 . . 3 (∀𝑝𝑋 {𝑥𝐴𝑝𝑥} ∈ Fin → (𝑃𝑋 → {𝑥𝐴𝑃𝑥} ∈ Fin))
83, 7syl 17 . 2 (𝐴 ∈ PtFin → (𝑃𝑋 → {𝑥𝐴𝑃𝑥} ∈ Fin))
98imp 408 1 ((𝐴 ∈ PtFin ∧ 𝑃𝑋) → {𝑥𝐴𝑃𝑥} ∈ Fin)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  wral 3063  {crab 3406   cuni 4864  Fincfn 8817  PtFincptfin 22782
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3064  df-rab 3407  df-v 3446  df-in 3916  df-ss 3926  df-uni 4865  df-ptfin 22785
This theorem is referenced by:  locfindis  22809  comppfsc  22811
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