Metamath Proof Explorer


Theorem mvrval2

Description: Value of the generating elements of the power series structure. (Contributed by Mario Carneiro, 7-Jan-2015)

Ref Expression
Hypotheses mvrfval.v 𝑉 = ( 𝐼 mVar 𝑅 )
mvrfval.d 𝐷 = { ∈ ( ℕ0m 𝐼 ) ∣ ( “ ℕ ) ∈ Fin }
mvrfval.z 0 = ( 0g𝑅 )
mvrfval.o 1 = ( 1r𝑅 )
mvrfval.i ( 𝜑𝐼𝑊 )
mvrfval.r ( 𝜑𝑅𝑌 )
mvrval.x ( 𝜑𝑋𝐼 )
mvrval2.f ( 𝜑𝐹𝐷 )
Assertion mvrval2 ( 𝜑 → ( ( 𝑉𝑋 ) ‘ 𝐹 ) = if ( 𝐹 = ( 𝑦𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) , 1 , 0 ) )

Proof

Step Hyp Ref Expression
1 mvrfval.v 𝑉 = ( 𝐼 mVar 𝑅 )
2 mvrfval.d 𝐷 = { ∈ ( ℕ0m 𝐼 ) ∣ ( “ ℕ ) ∈ Fin }
3 mvrfval.z 0 = ( 0g𝑅 )
4 mvrfval.o 1 = ( 1r𝑅 )
5 mvrfval.i ( 𝜑𝐼𝑊 )
6 mvrfval.r ( 𝜑𝑅𝑌 )
7 mvrval.x ( 𝜑𝑋𝐼 )
8 mvrval2.f ( 𝜑𝐹𝐷 )
9 1 2 3 4 5 6 7 mvrval ( 𝜑 → ( 𝑉𝑋 ) = ( 𝑓𝐷 ↦ if ( 𝑓 = ( 𝑦𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) , 1 , 0 ) ) )
10 9 fveq1d ( 𝜑 → ( ( 𝑉𝑋 ) ‘ 𝐹 ) = ( ( 𝑓𝐷 ↦ if ( 𝑓 = ( 𝑦𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) , 1 , 0 ) ) ‘ 𝐹 ) )
11 eqeq1 ( 𝑓 = 𝐹 → ( 𝑓 = ( 𝑦𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ↔ 𝐹 = ( 𝑦𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) ) )
12 11 ifbid ( 𝑓 = 𝐹 → if ( 𝑓 = ( 𝑦𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) , 1 , 0 ) = if ( 𝐹 = ( 𝑦𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) , 1 , 0 ) )
13 eqid ( 𝑓𝐷 ↦ if ( 𝑓 = ( 𝑦𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) , 1 , 0 ) ) = ( 𝑓𝐷 ↦ if ( 𝑓 = ( 𝑦𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) , 1 , 0 ) )
14 4 fvexi 1 ∈ V
15 3 fvexi 0 ∈ V
16 14 15 ifex if ( 𝐹 = ( 𝑦𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) , 1 , 0 ) ∈ V
17 12 13 16 fvmpt ( 𝐹𝐷 → ( ( 𝑓𝐷 ↦ if ( 𝑓 = ( 𝑦𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) , 1 , 0 ) ) ‘ 𝐹 ) = if ( 𝐹 = ( 𝑦𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) , 1 , 0 ) )
18 8 17 syl ( 𝜑 → ( ( 𝑓𝐷 ↦ if ( 𝑓 = ( 𝑦𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) , 1 , 0 ) ) ‘ 𝐹 ) = if ( 𝐹 = ( 𝑦𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) , 1 , 0 ) )
19 10 18 eqtrd ( 𝜑 → ( ( 𝑉𝑋 ) ‘ 𝐹 ) = if ( 𝐹 = ( 𝑦𝐼 ↦ if ( 𝑦 = 𝑋 , 1 , 0 ) ) , 1 , 0 ) )