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Metamath Proof Explorer

Theorem sumfc

Description: A lemma to facilitate conversions from the function form to the class-variable form of a sum. (Contributed by Mario Carneiro, 12-Aug-2013) (Revised by Mario Carneiro, 23-Apr-2014)

Ref Expression
Assertion sumfc Σ 𝑗 ∈ 𝐴 ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑗 ) = Σ 𝑘 ∈ 𝐴 𝐵

Proof

Step Hyp Ref Expression
1 eqid ⊢ ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) = ( 𝑘 ∈ 𝐴 ↦ 𝐵 )
2 1 fvmpt2i ⊢ ( 𝑘 ∈ 𝐴 → ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑘 ) = ( I ‘ 𝐵 ) )
3 2 sumeq2i ⊢ Σ 𝑘 ∈ 𝐴 ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑘 ) = Σ 𝑘 ∈ 𝐴 ( I ‘ 𝐵 )
4 fveq2 ⊢ ( 𝑗 = 𝑘 → ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑗 ) = ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑘 ) )
5 nffvmpt1 ⊢ Ⅎ 𝑘 ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑗 )
6 nfcv ⊢ Ⅎ 𝑗 ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑘 )
7 4 5 6 cbvsum ⊢ Σ 𝑗 ∈ 𝐴 ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑗 ) = Σ 𝑘 ∈ 𝐴 ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑘 )
8 sum2id ⊢ Σ 𝑘 ∈ 𝐴 𝐵 = Σ 𝑘 ∈ 𝐴 ( I ‘ 𝐵 )
9 3 7 8 3eqtr4i ⊢ Σ 𝑗 ∈ 𝐴 ( ( 𝑘 ∈ 𝐴 ↦ 𝐵 ) ‘ 𝑗 ) = Σ 𝑘 ∈ 𝐴 𝐵