Metamath Proof Explorer


Theorem fin34i

Description: Inference from isfin3-4 . (Contributed by Mario Carneiro, 17-May-2015)

Ref Expression
Assertion fin34i ( ( 𝐴 ∈ FinIII𝐺 : ω ⟶ 𝒫 𝐴 ∧ ∀ 𝑥 ∈ ω ( 𝐺𝑥 ) ⊆ ( 𝐺 ‘ suc 𝑥 ) ) → ran 𝐺 ∈ ran 𝐺 )

Proof

Step Hyp Ref Expression
1 eqid ( 𝑦 ∈ 𝒫 𝐴 ↦ ( 𝐴𝑦 ) ) = ( 𝑦 ∈ 𝒫 𝐴 ↦ ( 𝐴𝑦 ) )
2 1 isf34lem7 ( ( 𝐴 ∈ FinIII𝐺 : ω ⟶ 𝒫 𝐴 ∧ ∀ 𝑥 ∈ ω ( 𝐺𝑥 ) ⊆ ( 𝐺 ‘ suc 𝑥 ) ) → ran 𝐺 ∈ ran 𝐺 )