Theorem pwselbasr

Description: The reverse direction of pwselbasb : a function between the index and base set of a structure is an element of the structure power. (Contributed by SN, 29-Jul-2024)

	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 )
	pwselbasr.x	⊢ ( 𝜑 → 𝑋 : 𝐼 ⟶ 𝐵 )
Assertion	pwselbasr	⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )

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		pwselbasr.x	⊢ ( 𝜑 → 𝑋 : 𝐼 ⟶ 𝐵 )
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	mpbird	⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )