Theorem sumex

Description: A sum is a set. (Contributed by NM, 11-Dec-2005) (Revised by Mario Carneiro, 13-Jun-2019)

	Ref	Expression
Assertion	sumex	⊢ Σ 𝑘 ∈ 𝐴 𝐵 ∈ V

Proof

Step	Hyp	Ref	Expression
1		df-sum	⊢ Σ 𝑘 ∈ 𝐴 𝐵 = ( ℩ 𝑥 ( ∃ 𝑚 ∈ ℤ ( 𝐴 ⊆ ( ℤ≥ ‘ 𝑚 ) ∧ seq 𝑚 ( + , ( 𝑛 ∈ ℤ ↦ if ( 𝑛 ∈ 𝐴 , ⦋ 𝑛 / 𝑘 ⦌ 𝐵 , 0 ) ) ) ⇝ 𝑥 ) ∨ ∃ 𝑚 ∈ ℕ ∃ 𝑓 ( 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐴 ∧ 𝑥 = ( seq 1 ( + , ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐵 ) ) ‘ 𝑚 ) ) ) )
2		iotaex	⊢ ( ℩ 𝑥 ( ∃ 𝑚 ∈ ℤ ( 𝐴 ⊆ ( ℤ≥ ‘ 𝑚 ) ∧ seq 𝑚 ( + , ( 𝑛 ∈ ℤ ↦ if ( 𝑛 ∈ 𝐴 , ⦋ 𝑛 / 𝑘 ⦌ 𝐵 , 0 ) ) ) ⇝ 𝑥 ) ∨ ∃ 𝑚 ∈ ℕ ∃ 𝑓 ( 𝑓 : ( 1 ... 𝑚 ) –1-1-onto→ 𝐴 ∧ 𝑥 = ( seq 1 ( + , ( 𝑛 ∈ ℕ ↦ ⦋ ( 𝑓 ‘ 𝑛 ) / 𝑘 ⦌ 𝐵 ) ) ‘ 𝑚 ) ) ) ) ∈ V
3	1 2	eqeltri	⊢ Σ 𝑘 ∈ 𝐴 𝐵 ∈ V