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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 ⨉ 𝑠 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 ⨉ 𝑠 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 ∈ I Base R ⁡ x
8 6 7 eleqtrd ⊢ φ → T ∈ ⨉ x ∈ I Base R ⁡ x
9 ixpfn ⊢ T ∈ ⨉ x ∈ I Base R ⁡ x → T Fn I
10 8 9 syl ⊢ φ → T Fn I