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Theorem isptfin 21129
 Description: The statement "is a point-finite cover." (Contributed by Jeff Hankins, 21-Jan-2010.)
Hypothesis
Ref Expression
isptfin.1 𝑋 = 𝐴
Assertion
Ref Expression
isptfin (𝐴𝐵 → (𝐴 ∈ PtFin ↔ ∀𝑥𝑋 {𝑦𝐴𝑥𝑦} ∈ Fin))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝑋
Allowed substitution hints:   𝐵(𝑥,𝑦)   𝑋(𝑦)

Proof of Theorem isptfin
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 unieq 4380 . . . 4 (𝑎 = 𝐴 𝑎 = 𝐴)
2 isptfin.1 . . . 4 𝑋 = 𝐴
31, 2syl6eqr 2662 . . 3 (𝑎 = 𝐴 𝑎 = 𝑋)
4 rabeq 3166 . . . 4 (𝑎 = 𝐴 → {𝑦𝑎𝑥𝑦} = {𝑦𝐴𝑥𝑦})
54eleq1d 2672 . . 3 (𝑎 = 𝐴 → ({𝑦𝑎𝑥𝑦} ∈ Fin ↔ {𝑦𝐴𝑥𝑦} ∈ Fin))
63, 5raleqbidv 3129 . 2 (𝑎 = 𝐴 → (∀𝑥 𝑎{𝑦𝑎𝑥𝑦} ∈ Fin ↔ ∀𝑥𝑋 {𝑦𝐴𝑥𝑦} ∈ Fin))
7 df-ptfin 21119 . 2 PtFin = {𝑎 ∣ ∀𝑥 𝑎{𝑦𝑎𝑥𝑦} ∈ Fin}
86, 7elab2g 3322 1 (𝐴𝐵 → (𝐴 ∈ PtFin ↔ ∀𝑥𝑋 {𝑦𝐴𝑥𝑦} ∈ Fin))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {crab 2900  ∪ cuni 4372  Fincfn 7841  PtFincptfin 21116 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-uni 4373  df-ptfin 21119 This theorem is referenced by:  finptfin  21131  ptfinfin  21132  lfinpfin  21137
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