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Metamath Proof Explorer

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 ) )