Theorem elixpconstg

Description: Membership in an infinite Cartesian product of a constant B . (Contributed by Glauco Siliprandi, 8-Apr-2021)

	Ref	Expression
Assertion	elixpconstg	⊢ ( 𝐹 ∈ 𝑉 → ( 𝐹 ∈ X 𝑥 ∈ 𝐴 𝐵 ↔ 𝐹 : 𝐴 ⟶ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		ixpfn	⊢ ( 𝐹 ∈ X 𝑥 ∈ 𝐴 𝐵 → 𝐹 Fn 𝐴 )
2		elixp2	⊢ ( 𝐹 ∈ X 𝑥 ∈ 𝐴 𝐵 ↔ ( 𝐹 ∈ V ∧ 𝐹 Fn 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) ∈ 𝐵 ) )
3	2	simp3bi	⊢ ( 𝐹 ∈ X 𝑥 ∈ 𝐴 𝐵 → ∀ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) ∈ 𝐵 )
4		ffnfv	⊢ ( 𝐹 : 𝐴 ⟶ 𝐵 ↔ ( 𝐹 Fn 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) ∈ 𝐵 ) )
5	1 3 4	sylanbrc	⊢ ( 𝐹 ∈ X 𝑥 ∈ 𝐴 𝐵 → 𝐹 : 𝐴 ⟶ 𝐵 )
6		elex	⊢ ( 𝐹 ∈ 𝑉 → 𝐹 ∈ V )
7	6	adantr	⊢ ( ( 𝐹 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ 𝐵 ) → 𝐹 ∈ V )
8		ffn	⊢ ( 𝐹 : 𝐴 ⟶ 𝐵 → 𝐹 Fn 𝐴 )
9	8	adantl	⊢ ( ( 𝐹 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ 𝐵 ) → 𝐹 Fn 𝐴 )
10	4	simprbi	⊢ ( 𝐹 : 𝐴 ⟶ 𝐵 → ∀ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) ∈ 𝐵 )
11	10	adantl	⊢ ( ( 𝐹 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ 𝐵 ) → ∀ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) ∈ 𝐵 )
12	7 9 11 2	syl3anbrc	⊢ ( ( 𝐹 ∈ 𝑉 ∧ 𝐹 : 𝐴 ⟶ 𝐵 ) → 𝐹 ∈ X 𝑥 ∈ 𝐴 𝐵 )
13	12	ex	⊢ ( 𝐹 ∈ 𝑉 → ( 𝐹 : 𝐴 ⟶ 𝐵 → 𝐹 ∈ X 𝑥 ∈ 𝐴 𝐵 ) )
14	5 13	impbid2	⊢ ( 𝐹 ∈ 𝑉 → ( 𝐹 ∈ X 𝑥 ∈ 𝐴 𝐵 ↔ 𝐹 : 𝐴 ⟶ 𝐵 ) )