MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isfin1a Structured version   Visualization version   GIF version

Theorem isfin1a 10286
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 4611 . . 3 (𝑥 = 𝐴 → 𝒫 𝑥 = 𝒫 𝐴)
2 difeq1 4110 . . . . 5 (𝑥 = 𝐴 → (𝑥𝑦) = (𝐴𝑦))
32eleq1d 2812 . . . 4 (𝑥 = 𝐴 → ((𝑥𝑦) ∈ Fin ↔ (𝐴𝑦) ∈ Fin))
43orbi2d 912 . . 3 (𝑥 = 𝐴 → ((𝑦 ∈ Fin ∨ (𝑥𝑦) ∈ Fin) ↔ (𝑦 ∈ Fin ∨ (𝐴𝑦) ∈ Fin)))
51, 4raleqbidv 3336 . 2 (𝑥 = 𝐴 → (∀𝑦 ∈ 𝒫 𝑥(𝑦 ∈ Fin ∨ (𝑥𝑦) ∈ Fin) ↔ ∀𝑦 ∈ 𝒫 𝐴(𝑦 ∈ Fin ∨ (𝐴𝑦) ∈ Fin)))
6 df-fin1a 10279 . 2 FinIa = {𝑥 ∣ ∀𝑦 ∈ 𝒫 𝑥(𝑦 ∈ Fin ∨ (𝑥𝑦) ∈ Fin)}
75, 6elab2g 3665 1 (𝐴𝑉 → (𝐴 ∈ FinIa ↔ ∀𝑦 ∈ 𝒫 𝐴(𝑦 ∈ Fin ∨ (𝐴𝑦) ∈ Fin)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wo 844   = wceq 1533  wcel 2098  wral 3055  cdif 3940  𝒫 cpw 4597  Fincfn 8938  FinIacfin1a 10272
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rab 3427  df-v 3470  df-dif 3946  df-in 3950  df-ss 3960  df-pw 4599  df-fin1a 10279
This theorem is referenced by:  fin1ai  10287  fin11a  10377  enfin1ai  10378
  Copyright terms: Public domain W3C validator