Metamath Proof Explorer
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 ) |