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Theorem ptfinfin 23378
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 23375 . . . 4 (𝐴 ∈ PtFin → (𝐴 ∈ PtFin ↔ ∀𝑝𝑋 {𝑥𝐴𝑝𝑥} ∈ Fin))
32ibi 267 . . 3 (𝐴 ∈ PtFin → ∀𝑝𝑋 {𝑥𝐴𝑝𝑥} ∈ Fin)
4 eleq1 2815 . . . . . 6 (𝑝 = 𝑃 → (𝑝𝑥𝑃𝑥))
54rabbidv 3434 . . . . 5 (𝑝 = 𝑃 → {𝑥𝐴𝑝𝑥} = {𝑥𝐴𝑃𝑥})
65eleq1d 2812 . . . 4 (𝑝 = 𝑃 → ({𝑥𝐴𝑝𝑥} ∈ Fin ↔ {𝑥𝐴𝑃𝑥} ∈ Fin))
76rspccv 3603 . . 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 1533  wcel 2098  wral 3055  {crab 3426   cuni 4902  Fincfn 8941  PtFincptfin 23362
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rab 3427  df-v 3470  df-in 3950  df-ss 3960  df-uni 4903  df-ptfin 23365
This theorem is referenced by:  locfindis  23389  comppfsc  23391
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