Theorem seqfn

Description: The sequence builder function is a function. (Contributed by Mario Carneiro, 24-Jun-2013) (Revised by Mario Carneiro, 15-Sep-2013)

	Ref	Expression
Assertion	seqfn	⊢ ( 𝑀 ∈ ℤ → seq 𝑀 ( + , 𝐹 ) Fn ( ℤ≥ ‘ 𝑀 ) )

Proof

Step	Hyp	Ref	Expression
1		seqeq1	⊢ ( 𝑀 = if ( 𝑀 ∈ ℤ , 𝑀 , 0 ) → seq 𝑀 ( + , 𝐹 ) = seq if ( 𝑀 ∈ ℤ , 𝑀 , 0 ) ( + , 𝐹 ) )
2		fveq2	⊢ ( 𝑀 = if ( 𝑀 ∈ ℤ , 𝑀 , 0 ) → ( ℤ≥ ‘ 𝑀 ) = ( ℤ≥ ‘ if ( 𝑀 ∈ ℤ , 𝑀 , 0 ) ) )
3	1 2	fneq12d	⊢ ( 𝑀 = if ( 𝑀 ∈ ℤ , 𝑀 , 0 ) → ( seq 𝑀 ( + , 𝐹 ) Fn ( ℤ≥ ‘ 𝑀 ) ↔ seq if ( 𝑀 ∈ ℤ , 𝑀 , 0 ) ( + , 𝐹 ) Fn ( ℤ≥ ‘ if ( 𝑀 ∈ ℤ , 𝑀 , 0 ) ) ) )
4		0z	⊢ 0 ∈ ℤ
5	4	elimel	⊢ if ( 𝑀 ∈ ℤ , 𝑀 , 0 ) ∈ ℤ
6		eqid	⊢ ( rec ( ( 𝑥 ∈ V ↦ ( 𝑥 + 1 ) ) , if ( 𝑀 ∈ ℤ , 𝑀 , 0 ) ) ↾ ω ) = ( rec ( ( 𝑥 ∈ V ↦ ( 𝑥 + 1 ) ) , if ( 𝑀 ∈ ℤ , 𝑀 , 0 ) ) ↾ ω )
7		fvex	⊢ ( 𝐹 ‘ if ( 𝑀 ∈ ℤ , 𝑀 , 0 ) ) ∈ V
8		eqid	⊢ ( rec ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ⟨ ( 𝑥 + 1 ) , ( 𝑥 ( 𝑧 ∈ V , 𝑤 ∈ V ↦ ( 𝑤 + ( 𝐹 ‘ ( 𝑧 + 1 ) ) ) ) 𝑦 ) ⟩ ) , ⟨ if ( 𝑀 ∈ ℤ , 𝑀 , 0 ) , ( 𝐹 ‘ if ( 𝑀 ∈ ℤ , 𝑀 , 0 ) ) ⟩ ) ↾ ω ) = ( rec ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ⟨ ( 𝑥 + 1 ) , ( 𝑥 ( 𝑧 ∈ V , 𝑤 ∈ V ↦ ( 𝑤 + ( 𝐹 ‘ ( 𝑧 + 1 ) ) ) ) 𝑦 ) ⟩ ) , ⟨ if ( 𝑀 ∈ ℤ , 𝑀 , 0 ) , ( 𝐹 ‘ if ( 𝑀 ∈ ℤ , 𝑀 , 0 ) ) ⟩ ) ↾ ω )
9	8	seqval	⊢ seq if ( 𝑀 ∈ ℤ , 𝑀 , 0 ) ( + , 𝐹 ) = ran ( rec ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ⟨ ( 𝑥 + 1 ) , ( 𝑥 ( 𝑧 ∈ V , 𝑤 ∈ V ↦ ( 𝑤 + ( 𝐹 ‘ ( 𝑧 + 1 ) ) ) ) 𝑦 ) ⟩ ) , ⟨ if ( 𝑀 ∈ ℤ , 𝑀 , 0 ) , ( 𝐹 ‘ if ( 𝑀 ∈ ℤ , 𝑀 , 0 ) ) ⟩ ) ↾ ω )
10	5 6 7 8 9	uzrdgfni	⊢ seq if ( 𝑀 ∈ ℤ , 𝑀 , 0 ) ( + , 𝐹 ) Fn ( ℤ≥ ‘ if ( 𝑀 ∈ ℤ , 𝑀 , 0 ) )
11	3 10	dedth	⊢ ( 𝑀 ∈ ℤ → seq 𝑀 ( + , 𝐹 ) Fn ( ℤ≥ ‘ 𝑀 ) )