Theorem prdsbasmpt2

Description: A constructed tuple is a point in a structure product iff each coordinate is in the proper base set. (Contributed by Mario Carneiro, 3-Jul-2015) (Revised by Mario Carneiro, 13-Sep-2015)

	Ref	Expression
Hypotheses	prdsbasmpt2.y	⊢ 𝑌 = ( 𝑆 Xs ( 𝑥 ∈ 𝐼 ↦ 𝑅 ) )
	prdsbasmpt2.b	⊢ 𝐵 = ( Base ‘ 𝑌 )
	prdsbasmpt2.s	⊢ ( 𝜑 → 𝑆 ∈ 𝑉 )
	prdsbasmpt2.i	⊢ ( 𝜑 → 𝐼 ∈ 𝑊 )
	prdsbasmpt2.r	⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐼 𝑅 ∈ 𝑋 )
	prdsbasmpt2.k	⊢ 𝐾 = ( Base ‘ 𝑅 )
Assertion	prdsbasmpt2	⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐼 ↦ 𝑈 ) ∈ 𝐵 ↔ ∀ 𝑥 ∈ 𝐼 𝑈 ∈ 𝐾 ) )

Proof

Step	Hyp	Ref	Expression
1		prdsbasmpt2.y	⊢ 𝑌 = ( 𝑆 Xs ( 𝑥 ∈ 𝐼 ↦ 𝑅 ) )
2		prdsbasmpt2.b	⊢ 𝐵 = ( Base ‘ 𝑌 )
3		prdsbasmpt2.s	⊢ ( 𝜑 → 𝑆 ∈ 𝑉 )
4		prdsbasmpt2.i	⊢ ( 𝜑 → 𝐼 ∈ 𝑊 )
5		prdsbasmpt2.r	⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐼 𝑅 ∈ 𝑋 )
6		prdsbasmpt2.k	⊢ 𝐾 = ( Base ‘ 𝑅 )
7	1 2 3 4 5 6	prdsbas3	⊢ ( 𝜑 → 𝐵 = X 𝑥 ∈ 𝐼 𝐾 )
8	7	eleq2d	⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐼 ↦ 𝑈 ) ∈ 𝐵 ↔ ( 𝑥 ∈ 𝐼 ↦ 𝑈 ) ∈ X 𝑥 ∈ 𝐼 𝐾 ) )
9		mptelixpg	⊢ ( 𝐼 ∈ 𝑊 → ( ( 𝑥 ∈ 𝐼 ↦ 𝑈 ) ∈ X 𝑥 ∈ 𝐼 𝐾 ↔ ∀ 𝑥 ∈ 𝐼 𝑈 ∈ 𝐾 ) )
10	4 9	syl	⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐼 ↦ 𝑈 ) ∈ X 𝑥 ∈ 𝐼 𝐾 ↔ ∀ 𝑥 ∈ 𝐼 𝑈 ∈ 𝐾 ) )
11	8 10	bitrd	⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐼 ↦ 𝑈 ) ∈ 𝐵 ↔ ∀ 𝑥 ∈ 𝐼 𝑈 ∈ 𝐾 ) )