Theorem riotauni

Description: Restricted iota in terms of class union. (Contributed by NM, 11-Oct-2011)

	Ref	Expression
Assertion	riotauni	⊢ ( ∃! 𝑥 ∈ 𝐴 𝜑 → ( ℩ 𝑥 ∈ 𝐴 𝜑 ) = ∪ { 𝑥 ∈ 𝐴 ∣ 𝜑 } )

Proof

Step	Hyp	Ref	Expression
1		df-reu	⊢ ( ∃! 𝑥 ∈ 𝐴 𝜑 ↔ ∃! 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) )
2		iotauni	⊢ ( ∃! 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) → ( ℩ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ) = ∪ { 𝑥 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) } )
3	1 2	sylbi	⊢ ( ∃! 𝑥 ∈ 𝐴 𝜑 → ( ℩ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ) = ∪ { 𝑥 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) } )
4		df-riota	⊢ ( ℩ 𝑥 ∈ 𝐴 𝜑 ) = ( ℩ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) )
5		df-rab	⊢ { 𝑥 ∈ 𝐴 ∣ 𝜑 } = { 𝑥 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) }
6	5	unieqi	⊢ ∪ { 𝑥 ∈ 𝐴 ∣ 𝜑 } = ∪ { 𝑥 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) }
7	3 4 6	3eqtr4g	⊢ ( ∃! 𝑥 ∈ 𝐴 𝜑 → ( ℩ 𝑥 ∈ 𝐴 𝜑 ) = ∪ { 𝑥 ∈ 𝐴 ∣ 𝜑 } )