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Metamath Proof Explorer

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 ) )