Step |
Hyp |
Ref |
Expression |
1 |
|
mvrfval.v |
⊢ 𝑉 = ( 𝐼 mVar 𝑅 ) |
2 |
|
mvrfval.d |
⊢ 𝐷 = { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin } |
3 |
|
mvrfval.z |
⊢ 0 = ( 0g ‘ 𝑅 ) |
4 |
|
mvrfval.o |
⊢ 1 = ( 1r ‘ 𝑅 ) |
5 |
|
mvrfval.i |
⊢ ( 𝜑 → 𝐼 ∈ 𝑊 ) |
6 |
|
mvrfval.r |
⊢ ( 𝜑 → 𝑅 ∈ 𝑌 ) |
7 |
|
mvrval.x |
⊢ ( 𝜑 → 𝑋 ∈ 𝐼 ) |
8 |
|
mvrval2.f |
⊢ ( 𝜑 → 𝐹 ∈ 𝐷 ) |
9 |
1 2 3 4 5 6 7
|
mvrval |
⊢ ( 𝜑 → ( 𝑉 ‘ 𝑋 ) = ( 𝑓 ∈ 𝐷 ↦ if ( 𝑓 = ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) , 1 , 0 ) ) ) |
10 |
9
|
fveq1d |
⊢ ( 𝜑 → ( ( 𝑉 ‘ 𝑋 ) ‘ 𝐹 ) = ( ( 𝑓 ∈ 𝐷 ↦ if ( 𝑓 = ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) , 1 , 0 ) ) ‘ 𝐹 ) ) |
11 |
|
eqeq1 |
⊢ ( 𝑓 = 𝐹 → ( 𝑓 = ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ↔ 𝐹 = ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) ) |
12 |
11
|
ifbid |
⊢ ( 𝑓 = 𝐹 → if ( 𝑓 = ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) , 1 , 0 ) = if ( 𝐹 = ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) , 1 , 0 ) ) |
13 |
|
eqid |
⊢ ( 𝑓 ∈ 𝐷 ↦ if ( 𝑓 = ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) , 1 , 0 ) ) = ( 𝑓 ∈ 𝐷 ↦ if ( 𝑓 = ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) , 1 , 0 ) ) |
14 |
4
|
fvexi |
⊢ 1 ∈ V |
15 |
3
|
fvexi |
⊢ 0 ∈ V |
16 |
14 15
|
ifex |
⊢ if ( 𝐹 = ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) , 1 , 0 ) ∈ V |
17 |
12 13 16
|
fvmpt |
⊢ ( 𝐹 ∈ 𝐷 → ( ( 𝑓 ∈ 𝐷 ↦ if ( 𝑓 = ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) , 1 , 0 ) ) ‘ 𝐹 ) = if ( 𝐹 = ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) , 1 , 0 ) ) |
18 |
8 17
|
syl |
⊢ ( 𝜑 → ( ( 𝑓 ∈ 𝐷 ↦ if ( 𝑓 = ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) , 1 , 0 ) ) ‘ 𝐹 ) = if ( 𝐹 = ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) , 1 , 0 ) ) |
19 |
10 18
|
eqtrd |
⊢ ( 𝜑 → ( ( 𝑉 ‘ 𝑋 ) ‘ 𝐹 ) = if ( 𝐹 = ( 𝑦 ∈ 𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) , 1 , 0 ) ) |