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Theorem isfin1a 8997
 Description: Definition of a Ia-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)
Assertion
Ref Expression
isfin1a (𝐴𝑉 → (𝐴 ∈ FinIa ↔ ∀𝑦 ∈ 𝒫 𝐴(𝑦 ∈ Fin ∨ (𝐴𝑦) ∈ Fin)))
Distinct variable group:   𝑦,𝐴
Allowed substitution hint:   𝑉(𝑦)

Proof of Theorem isfin1a
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 pweq 4111 . . 3 (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴)
2 difeq1 3683 . . . . 5 (𝑥 = 𝐴 → (𝑥𝑦) = (𝐴𝑦))
32eleq1d 2672 . . . 4 (𝑥 = 𝐴 → ((𝑥𝑦) ∈ Fin ↔ (𝐴𝑦) ∈ Fin))
43orbi2d 734 . . 3 (𝑥 = 𝐴 → ((𝑦 ∈ Fin ∨ (𝑥𝑦) ∈ Fin) ↔ (𝑦 ∈ Fin ∨ (𝐴𝑦) ∈ Fin)))
51, 4raleqbidv 3129 . 2 (𝑥 = 𝐴 → (∀𝑦 ∈ 𝒫 𝑥(𝑦 ∈ Fin ∨ (𝑥𝑦) ∈ Fin) ↔ ∀𝑦 ∈ 𝒫 𝐴(𝑦 ∈ Fin ∨ (𝐴𝑦) ∈ Fin)))
6 df-fin1a 8990 . 2 FinIa = {𝑥 ∣ ∀𝑦 ∈ 𝒫 𝑥(𝑦 ∈ Fin ∨ (𝑥𝑦) ∈ Fin)}
75, 6elab2g 3322 1 (𝐴𝑉 → (𝐴 ∈ FinIa ↔ ∀𝑦 ∈ 𝒫 𝐴(𝑦 ∈ Fin ∨ (𝐴𝑦) ∈ Fin)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∨ wo 382   = wceq 1475   ∈ wcel 1977  ∀wral 2896   ∖ cdif 3537  𝒫 cpw 4108  Fincfn 7841  FinIacfin1a 8983 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-rab 2905  df-v 3175  df-dif 3543  df-in 3547  df-ss 3554  df-pw 4110  df-fin1a 8990 This theorem is referenced by:  fin1ai  8998  fin11a  9088  enfin1ai  9089
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