Theorem isfin1a 10216

Description: Definition of a Ia-finite set. (Contributed by Stefan O'Rear, 16-May-2015.)

Assertion

Ref	Expression
isfin1a	⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ FinIa ↔ ∀𝑦 ∈ 𝒫 𝐴(𝑦 ∈ Fin ∨ (𝐴 ∖ 𝑦) ∈ Fin)))

Distinct variable group:   𝑦,𝐴

Allowed substitution hint:   𝑉(𝑦)

Proof of Theorem isfin1a

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		pweq 4570	. . 3 ⊢ (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴)
2		difeq1 4073	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∖ 𝑦) = (𝐴 ∖ 𝑦))
3	2	eleq1d 2822	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 ∖ 𝑦) ∈ Fin ↔ (𝐴 ∖ 𝑦) ∈ Fin))
4	3	orbi2d 916	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑦 ∈ Fin ∨ (𝑥 ∖ 𝑦) ∈ Fin) ↔ (𝑦 ∈ Fin ∨ (𝐴 ∖ 𝑦) ∈ Fin)))
5	1, 4	raleqbidv 3318	. 2 ⊢ (𝑥 = 𝐴 → (∀𝑦 ∈ 𝒫 𝑥(𝑦 ∈ Fin ∨ (𝑥 ∖ 𝑦) ∈ Fin) ↔ ∀𝑦 ∈ 𝒫 𝐴(𝑦 ∈ Fin ∨ (𝐴 ∖ 𝑦) ∈ Fin)))
6		df-fin1a 10209	. 2 ⊢ FinIa = {𝑥 ∣ ∀𝑦 ∈ 𝒫 𝑥(𝑦 ∈ Fin ∨ (𝑥 ∖ 𝑦) ∈ Fin)}
7	5, 6	elab2g 3637	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ FinIa ↔ ∀𝑦 ∈ 𝒫 𝐴(𝑦 ∈ Fin ∨ (𝐴 ∖ 𝑦) ∈ Fin)))

Syntax hints:   → wi 4   ↔ wb 206   ∨ wo 848   = wceq 1542   ∈ wcel 2114  ∀wral 3052   ∖ cdif 3900  𝒫 cpw 4556  Fincfn 8897  Fin^Iacfin1a 10202

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 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rab 3402  df-v 3444  df-dif 3906  df-ss 3920  df-pw 4558  df-fin1a 10209

This theorem is referenced by:  fin1ai  10217  fin11a  10307  enfin1ai  10308