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Metamath Proof Explorer

Theorem prdsbasmpt

Description: A constructed tuple is a point in a structure product iff each coordinate is in the proper base set. (Contributed by Stefan O'Rear, 10-Jan-2015)

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

Proof

Step Hyp Ref Expression
1 prdsbasmpt.y ⊢ 𝑌 = ( 𝑆 Xs 𝑅 )
2 prdsbasmpt.b ⊢ 𝐵 = ( Base ‘ 𝑌 )
3 prdsbasmpt.s ⊢ ( 𝜑 → 𝑆 ∈ 𝑉 )
4 prdsbasmpt.i ⊢ ( 𝜑 → 𝐼 ∈ 𝑊 )
5 prdsbasmpt.r ⊢ ( 𝜑 → 𝑅 Fn 𝐼 )
6 1 2 3 4 5 prdsbas2 ⊢ ( 𝜑 → 𝐵 = X 𝑥 ∈ 𝐼 ( Base ‘ ( 𝑅 ‘ 𝑥 ) ) )
7 6 eleq2d ⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐼 ↦ 𝑈 ) ∈ 𝐵 ↔ ( 𝑥 ∈ 𝐼 ↦ 𝑈 ) ∈ X 𝑥 ∈ 𝐼 ( Base ‘ ( 𝑅 ‘ 𝑥 ) ) ) )
8 mptelixpg ⊢ ( 𝐼 ∈ 𝑊 → ( ( 𝑥 ∈ 𝐼 ↦ 𝑈 ) ∈ X 𝑥 ∈ 𝐼 ( Base ‘ ( 𝑅 ‘ 𝑥 ) ) ↔ ∀ 𝑥 ∈ 𝐼 𝑈 ∈ ( Base ‘ ( 𝑅 ‘ 𝑥 ) ) ) )
9 4 8 syl ⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐼 ↦ 𝑈 ) ∈ X 𝑥 ∈ 𝐼 ( Base ‘ ( 𝑅 ‘ 𝑥 ) ) ↔ ∀ 𝑥 ∈ 𝐼 𝑈 ∈ ( Base ‘ ( 𝑅 ‘ 𝑥 ) ) ) )
10 7 9 bitrd ⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐼 ↦ 𝑈 ) ∈ 𝐵 ↔ ∀ 𝑥 ∈ 𝐼 𝑈 ∈ ( Base ‘ ( 𝑅 ‘ 𝑥 ) ) ) )