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Theorem ptfinfin 21815
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 21812 . . . 4 (𝐴 ∈ PtFin → (𝐴 ∈ PtFin ↔ ∀𝑝𝑋 {𝑥𝐴𝑝𝑥} ∈ Fin))
32ibi 268 . . 3 (𝐴 ∈ PtFin → ∀𝑝𝑋 {𝑥𝐴𝑝𝑥} ∈ Fin)
4 eleq1 2872 . . . . . 6 (𝑝 = 𝑃 → (𝑝𝑥𝑃𝑥))
54rabbidv 3428 . . . . 5 (𝑝 = 𝑃 → {𝑥𝐴𝑝𝑥} = {𝑥𝐴𝑃𝑥})
65eleq1d 2869 . . . 4 (𝑝 = 𝑃 → ({𝑥𝐴𝑝𝑥} ∈ Fin ↔ {𝑥𝐴𝑃𝑥} ∈ Fin))
76rspccv 3558 . . 3 (∀𝑝𝑋 {𝑥𝐴𝑝𝑥} ∈ Fin → (𝑃𝑋 → {𝑥𝐴𝑃𝑥} ∈ Fin))
83, 7syl 17 . 2 (𝐴 ∈ PtFin → (𝑃𝑋 → {𝑥𝐴𝑃𝑥} ∈ Fin))
98imp 407 1 ((𝐴 ∈ PtFin ∧ 𝑃𝑋) → {𝑥𝐴𝑃𝑥} ∈ Fin)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1525  wcel 2083  wral 3107  {crab 3111   cuni 4751  Fincfn 8364  PtFincptfin 21799
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-ext 2771
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ral 3112  df-rex 3113  df-rab 3116  df-v 3442  df-uni 4752  df-ptfin 21802
This theorem is referenced by:  locfindis  21826  comppfsc  21828
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