Theorem pwselbas

Description: An element of a structure power is a function from the index set to the base set of the structure. (Contributed by Mario Carneiro, 11-Jan-2015) (Revised by Mario Carneiro, 5-Jun-2015)

	Ref	Expression
Hypotheses	pwsbas.y	⊢ 𝑌 = ( 𝑅 ↑s 𝐼 )
	pwsbas.f	⊢ 𝐵 = ( Base ‘ 𝑅 )
	pwselbas.v	⊢ 𝑉 = ( Base ‘ 𝑌 )
	pwselbas.r	⊢ ( 𝜑 → 𝑅 ∈ 𝑊 )
	pwselbas.i	⊢ ( 𝜑 → 𝐼 ∈ 𝑍 )
	pwselbas.x	⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
Assertion	pwselbas	⊢ ( 𝜑 → 𝑋 : 𝐼 ⟶ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		pwsbas.y	⊢ 𝑌 = ( 𝑅 ↑s 𝐼 )
2		pwsbas.f	⊢ 𝐵 = ( Base ‘ 𝑅 )
3		pwselbas.v	⊢ 𝑉 = ( Base ‘ 𝑌 )
4		pwselbas.r	⊢ ( 𝜑 → 𝑅 ∈ 𝑊 )
5		pwselbas.i	⊢ ( 𝜑 → 𝐼 ∈ 𝑍 )
6		pwselbas.x	⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
7	1 2 3	pwselbasb	⊢ ( ( 𝑅 ∈ 𝑊 ∧ 𝐼 ∈ 𝑍 ) → ( 𝑋 ∈ 𝑉 ↔ 𝑋 : 𝐼 ⟶ 𝐵 ) )
8	4 5 7	syl2anc	⊢ ( 𝜑 → ( 𝑋 ∈ 𝑉 ↔ 𝑋 : 𝐼 ⟶ 𝐵 ) )
9	6 8	mpbid	⊢ ( 𝜑 → 𝑋 : 𝐼 ⟶ 𝐵 )