Theorem riotacl2

Description: Membership law for "the unique element in A such that ph ". (Contributed by NM, 21-Aug-2011) (Revised by Mario Carneiro, 23-Dec-2016)

	Ref	Expression
Assertion	riotacl2	⊢ ( ∃! 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		iotacl	⊢ ( ∃! 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	3 4 5	3eltr4g	⊢ ( ∃! 𝑥 ∈ 𝐴 𝜑 → ( ℩ 𝑥 ∈ 𝐴 𝜑 ) ∈ { 𝑥 ∈ 𝐴 ∣ 𝜑 } )