Theorem snriota

Description: A restricted class abstraction with a unique member can be expressed as a singleton. (Contributed by NM, 30-May-2006)

	Ref	Expression
Assertion	snriota	⊢ ( ∃! 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		sniota	⊢ ( ∃! 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-rab	⊢ { 𝑥 ∈ 𝐴 ∣ 𝜑 } = { 𝑥 ∣ ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) }
5		df-riota	⊢ ( ℩ 𝑥 ∈ 𝐴 𝜑 ) = ( ℩ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) )
6	5	sneqi	⊢ { ( ℩ 𝑥 ∈ 𝐴 𝜑 ) } = { ( ℩ 𝑥 ( 𝑥 ∈ 𝐴 ∧ 𝜑 ) ) }
7	3 4 6	3eqtr4g	⊢ ( ∃! 𝑥 ∈ 𝐴 𝜑 → { 𝑥 ∈ 𝐴 ∣ 𝜑 } = { ( ℩ 𝑥 ∈ 𝐴 𝜑 ) } )