MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ptfinfin Structured version   Visualization version   GIF version

Theorem ptfinfin 23527
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 23524 . . . 4 (𝐴 ∈ PtFin → (𝐴 ∈ PtFin ↔ ∀𝑝𝑋 {𝑥𝐴𝑝𝑥} ∈ Fin))
32ibi 267 . . 3 (𝐴 ∈ PtFin → ∀𝑝𝑋 {𝑥𝐴𝑝𝑥} ∈ Fin)
4 eleq1 2829 . . . . . 6 (𝑝 = 𝑃 → (𝑝𝑥𝑃𝑥))
54rabbidv 3444 . . . . 5 (𝑝 = 𝑃 → {𝑥𝐴𝑝𝑥} = {𝑥𝐴𝑃𝑥})
65eleq1d 2826 . . . 4 (𝑝 = 𝑃 → ({𝑥𝐴𝑝𝑥} ∈ Fin ↔ {𝑥𝐴𝑃𝑥} ∈ Fin))
76rspccv 3619 . . 3 (∀𝑝𝑋 {𝑥𝐴𝑝𝑥} ∈ Fin → (𝑃𝑋 → {𝑥𝐴𝑃𝑥} ∈ Fin))
83, 7syl 17 . 2 (𝐴 ∈ PtFin → (𝑃𝑋 → {𝑥𝐴𝑃𝑥} ∈ Fin))
98imp 406 1 ((𝐴 ∈ PtFin ∧ 𝑃𝑋) → {𝑥𝐴𝑃𝑥} ∈ Fin)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  wral 3061  {crab 3436   cuni 4907  Fincfn 8985  PtFincptfin 23511
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rab 3437  df-v 3482  df-ss 3968  df-uni 4908  df-ptfin 23514
This theorem is referenced by:  locfindis  23538  comppfsc  23540
  Copyright terms: Public domain W3C validator