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Metamath Proof Explorer

Theorem prdsbascl

Description: An element of the base has projections closed in the factors. (Contributed by Mario Carneiro, 27-Aug-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 )
prdsbascl.f ⊢ ( 𝜑 → 𝐹 ∈ 𝐵 )
Assertion prdsbascl ( 𝜑 → ∀ 𝑥 ∈ 𝐼 ( 𝐹 ‘ 𝑥 ) ∈ 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 prdsbascl.f ⊢ ( 𝜑 → 𝐹 ∈ 𝐵 )
8 eqid ⊢ ( 𝑥 ∈ 𝐼 ↦ 𝑅 ) = ( 𝑥 ∈ 𝐼 ↦ 𝑅 )
9 8 fnmpt ⊢ ( ∀ 𝑥 ∈ 𝐼 𝑅 ∈ 𝑋 → ( 𝑥 ∈ 𝐼 ↦ 𝑅 ) Fn 𝐼 )
10 5 9 syl ⊢ ( 𝜑 → ( 𝑥 ∈ 𝐼 ↦ 𝑅 ) Fn 𝐼 )
11 1 2 3 4 10 7 prdsbasfn ⊢ ( 𝜑 → 𝐹 Fn 𝐼 )
12 dffn5 ⊢ ( 𝐹 Fn 𝐼 ↔ 𝐹 = ( 𝑥 ∈ 𝐼 ↦ ( 𝐹 ‘ 𝑥 ) ) )
13 11 12 sylib ⊢ ( 𝜑 → 𝐹 = ( 𝑥 ∈ 𝐼 ↦ ( 𝐹 ‘ 𝑥 ) ) )
14 13 7 eqeltrrd ⊢ ( 𝜑 → ( 𝑥 ∈ 𝐼 ↦ ( 𝐹 ‘ 𝑥 ) ) ∈ 𝐵 )
15 1 2 3 4 5 6 prdsbasmpt2 ⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐼 ↦ ( 𝐹 ‘ 𝑥 ) ) ∈ 𝐵 ↔ ∀ 𝑥 ∈ 𝐼 ( 𝐹 ‘ 𝑥 ) ∈ 𝐾 ) )
16 14 15 mpbid ⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐼 ( 𝐹 ‘ 𝑥 ) ∈ 𝐾 )