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Metamath Proof Explorer

Theorem ixpconst

Description: Infinite Cartesian product of a constant B . (Contributed by NM, 28-Sep-2006)

Ref Expression
Hypotheses ixpconst.1 ⊢ 𝐴 ∈ V
ixpconst.2 ⊢ 𝐵 ∈ V
Assertion ixpconst X 𝑥 ∈ 𝐴 𝐵 = ( 𝐵 ↑m 𝐴 )

Proof

Step Hyp Ref Expression
1 ixpconst.1 ⊢ 𝐴 ∈ V
2 ixpconst.2 ⊢ 𝐵 ∈ V
3 ixpconstg ⊢ ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) → X 𝑥 ∈ 𝐴 𝐵 = ( 𝐵 ↑m 𝐴 ) )
4 1 2 3 mp2an ⊢ X 𝑥 ∈ 𝐴 𝐵 = ( 𝐵 ↑m 𝐴 )