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Metamath Proof Explorer

Theorem prdsbasfn

Description: Points in the structure product are functions; use this with dffn5 to establish equalities. (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 )
prdsbasmpt.t ⊢ ( 𝜑 → 𝑇 ∈ 𝐵 )
Assertion prdsbasfn ( 𝜑 → 𝑇 Fn 𝐼 )

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 prdsbasmpt.t ⊢ ( 𝜑 → 𝑇 ∈ 𝐵 )
7 1 2 3 4 5 prdsbas2 ⊢ ( 𝜑 → 𝐵 = X 𝑥 ∈ 𝐼 ( Base ‘ ( 𝑅 ‘ 𝑥 ) ) )
8 6 7 eleqtrd ⊢ ( 𝜑 → 𝑇 ∈ X 𝑥 ∈ 𝐼 ( Base ‘ ( 𝑅 ‘ 𝑥 ) ) )
9 ixpfn ⊢ ( 𝑇 ∈ X 𝑥 ∈ 𝐼 ( Base ‘ ( 𝑅 ‘ 𝑥 ) ) → 𝑇 Fn 𝐼 )
10 8 9 syl ⊢ ( 𝜑 → 𝑇 Fn 𝐼 )