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Theorem isfin1a 9397
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 4352 . . 3 (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴)
2 difeq1 3918 . . . . 5 (𝑥 = 𝐴 → (𝑥𝑦) = (𝐴𝑦))
32eleq1d 2868 . . . 4 (𝑥 = 𝐴 → ((𝑥𝑦) ∈ Fin ↔ (𝐴𝑦) ∈ Fin))
43orbi2d 930 . . 3 (𝑥 = 𝐴 → ((𝑦 ∈ Fin ∨ (𝑥𝑦) ∈ Fin) ↔ (𝑦 ∈ Fin ∨ (𝐴𝑦) ∈ Fin)))
51, 4raleqbidv 3339 . 2 (𝑥 = 𝐴 → (∀𝑦 ∈ 𝒫 𝑥(𝑦 ∈ Fin ∨ (𝑥𝑦) ∈ Fin) ↔ ∀𝑦 ∈ 𝒫 𝐴(𝑦 ∈ Fin ∨ (𝐴𝑦) ∈ Fin)))
6 df-fin1a 9390 . 2 FinIa = {𝑥 ∣ ∀𝑦 ∈ 𝒫 𝑥(𝑦 ∈ Fin ∨ (𝑥𝑦) ∈ Fin)}
75, 6elab2g 3546 1 (𝐴𝑉 → (𝐴 ∈ FinIa ↔ ∀𝑦 ∈ 𝒫 𝐴(𝑦 ∈ Fin ∨ (𝐴𝑦) ∈ Fin)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wo 865   = wceq 1637  wcel 2156  wral 3094  cdif 3764  𝒫 cpw 4349  Fincfn 8190  FinIacfin1a 9383
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2782
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-clab 2791  df-cleq 2797  df-clel 2800  df-nfc 2935  df-ral 3099  df-rab 3103  df-v 3391  df-dif 3770  df-in 3774  df-ss 3781  df-pw 4351  df-fin1a 9390
This theorem is referenced by:  fin1ai  9398  fin11a  9488  enfin1ai  9489
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