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Theorem isptfin 22127
 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 4835 . . . 4 (𝑎 = 𝐴 𝑎 = 𝐴)
2 isptfin.1 . . . 4 𝑋 = 𝐴
31, 2syl6eqr 2877 . . 3 (𝑎 = 𝐴 𝑎 = 𝑋)
4 rabeq 3469 . . . 4 (𝑎 = 𝐴 → {𝑦𝑎𝑥𝑦} = {𝑦𝐴𝑥𝑦})
54eleq1d 2900 . . 3 (𝑎 = 𝐴 → ({𝑦𝑎𝑥𝑦} ∈ Fin ↔ {𝑦𝐴𝑥𝑦} ∈ Fin))
63, 5raleqbidv 3392 . 2 (𝑎 = 𝐴 → (∀𝑥 𝑎{𝑦𝑎𝑥𝑦} ∈ Fin ↔ ∀𝑥𝑋 {𝑦𝐴𝑥𝑦} ∈ Fin))
7 df-ptfin 22117 . 2 PtFin = {𝑎 ∣ ∀𝑥 𝑎{𝑦𝑎𝑥𝑦} ∈ Fin}
86, 7elab2g 3654 1 (𝐴𝐵 → (𝐴 ∈ PtFin ↔ ∀𝑥𝑋 {𝑦𝐴𝑥𝑦} ∈ Fin))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   = wceq 1538   ∈ wcel 2115  ∀wral 3133  {crab 3137  ∪ cuni 4824  Fincfn 8505  PtFincptfin 22114 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rab 3142  df-v 3482  df-in 3926  df-ss 3936  df-uni 4825  df-ptfin 22117 This theorem is referenced by:  finptfin  22129  ptfinfin  22130  lfinpfin  22135
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