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 ( 𝜑𝑆𝑉 )
prdsbasmpt.i ( 𝜑𝐼𝑊 )
prdsbasmpt.r ( 𝜑𝑅 Fn 𝐼 )
prdsbasmpt.t ( 𝜑𝑇𝐵 )
Assertion prdsbasfn ( 𝜑𝑇 Fn 𝐼 )

Proof

Step Hyp Ref Expression
1 prdsbasmpt.y 𝑌 = ( 𝑆 Xs 𝑅 )
2 prdsbasmpt.b 𝐵 = ( Base ‘ 𝑌 )
3 prdsbasmpt.s ( 𝜑𝑆𝑉 )
4 prdsbasmpt.i ( 𝜑𝐼𝑊 )
5 prdsbasmpt.r ( 𝜑𝑅 Fn 𝐼 )
6 prdsbasmpt.t ( 𝜑𝑇𝐵 )
7 1 2 3 4 5 prdsbas2 ( 𝜑𝐵 = X 𝑥𝐼 ( Base ‘ ( 𝑅𝑥 ) ) )
8 6 7 eleqtrd ( 𝜑𝑇X 𝑥𝐼 ( Base ‘ ( 𝑅𝑥 ) ) )
9 ixpfn ( 𝑇X 𝑥𝐼 ( Base ‘ ( 𝑅𝑥 ) ) → 𝑇 Fn 𝐼 )
10 8 9 syl ( 𝜑𝑇 Fn 𝐼 )