Theorem seqex

Description: Existence of the sequence builder operation. (Contributed by Mario Carneiro, 4-Sep-2013)

	Ref	Expression
Assertion	seqex	⊢ seq 𝑀 ( + , 𝐹 ) ∈ V

Proof

Step	Hyp	Ref	Expression
1		df-seq	⊢ seq 𝑀 ( + , 𝐹 ) = ( rec ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ⟨ ( 𝑥 + 1 ) , ( 𝑦 + ( 𝐹 ‘ ( 𝑥 + 1 ) ) ) ⟩ ) , ⟨ 𝑀 , ( 𝐹 ‘ 𝑀 ) ⟩ ) “ ω )
2		rdgfun	⊢ Fun rec ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ⟨ ( 𝑥 + 1 ) , ( 𝑦 + ( 𝐹 ‘ ( 𝑥 + 1 ) ) ) ⟩ ) , ⟨ 𝑀 , ( 𝐹 ‘ 𝑀 ) ⟩ )
3		omex	⊢ ω ∈ V
4		funimaexg	⊢ ( ( Fun rec ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ⟨ ( 𝑥 + 1 ) , ( 𝑦 + ( 𝐹 ‘ ( 𝑥 + 1 ) ) ) ⟩ ) , ⟨ 𝑀 , ( 𝐹 ‘ 𝑀 ) ⟩ ) ∧ ω ∈ V ) → ( rec ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ⟨ ( 𝑥 + 1 ) , ( 𝑦 + ( 𝐹 ‘ ( 𝑥 + 1 ) ) ) ⟩ ) , ⟨ 𝑀 , ( 𝐹 ‘ 𝑀 ) ⟩ ) “ ω ) ∈ V )
5	2 3 4	mp2an	⊢ ( rec ( ( 𝑥 ∈ V , 𝑦 ∈ V ↦ ⟨ ( 𝑥 + 1 ) , ( 𝑦 + ( 𝐹 ‘ ( 𝑥 + 1 ) ) ) ⟩ ) , ⟨ 𝑀 , ( 𝐹 ‘ 𝑀 ) ⟩ ) “ ω ) ∈ V
6	1 5	eqeltri	⊢ seq 𝑀 ( + , 𝐹 ) ∈ V