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Theorem isptfin 22819
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 4874 . . . 4 (𝑎 = 𝐴 𝑎 = 𝐴)
2 isptfin.1 . . . 4 𝑋 = 𝐴
31, 2eqtr4di 2795 . . 3 (𝑎 = 𝐴 𝑎 = 𝑋)
4 rabeq 3419 . . . 4 (𝑎 = 𝐴 → {𝑦𝑎𝑥𝑦} = {𝑦𝐴𝑥𝑦})
54eleq1d 2822 . . 3 (𝑎 = 𝐴 → ({𝑦𝑎𝑥𝑦} ∈ Fin ↔ {𝑦𝐴𝑥𝑦} ∈ Fin))
63, 5raleqbidv 3317 . 2 (𝑎 = 𝐴 → (∀𝑥 𝑎{𝑦𝑎𝑥𝑦} ∈ Fin ↔ ∀𝑥𝑋 {𝑦𝐴𝑥𝑦} ∈ Fin))
7 df-ptfin 22809 . 2 PtFin = {𝑎 ∣ ∀𝑥 𝑎{𝑦𝑎𝑥𝑦} ∈ Fin}
86, 7elab2g 3630 1 (𝐴𝐵 → (𝐴 ∈ PtFin ↔ ∀𝑥𝑋 {𝑦𝐴𝑥𝑦} ∈ Fin))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1541  wcel 2106  wral 3062  {crab 3405   cuni 4863  Fincfn 8841  PtFincptfin 22806
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3063  df-rab 3406  df-v 3445  df-in 3915  df-ss 3925  df-uni 4864  df-ptfin 22809
This theorem is referenced by:  finptfin  22821  ptfinfin  22822  lfinpfin  22827
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