Theorem ssrabeq

Description: If the restricting class of a restricted class abstraction is a subset of this restricted class abstraction, it is equal to this restricted class abstraction. (Contributed by Alexander van der Vekens, 31-Dec-2017)

	Ref	Expression
Assertion	ssrabeq	⊢ ( 𝑉 ⊆ { 𝑥 ∈ 𝑉 ∣ 𝜑 } ↔ 𝑉 = { 𝑥 ∈ 𝑉 ∣ 𝜑 } )

Proof

Step	Hyp	Ref	Expression
1		ssrab2	⊢ { 𝑥 ∈ 𝑉 ∣ 𝜑 } ⊆ 𝑉
2	1	biantru	⊢ ( 𝑉 ⊆ { 𝑥 ∈ 𝑉 ∣ 𝜑 } ↔ ( 𝑉 ⊆ { 𝑥 ∈ 𝑉 ∣ 𝜑 } ∧ { 𝑥 ∈ 𝑉 ∣ 𝜑 } ⊆ 𝑉 ) )
3		eqss	⊢ ( 𝑉 = { 𝑥 ∈ 𝑉 ∣ 𝜑 } ↔ ( 𝑉 ⊆ { 𝑥 ∈ 𝑉 ∣ 𝜑 } ∧ { 𝑥 ∈ 𝑉 ∣ 𝜑 } ⊆ 𝑉 ) )
4	2 3	bitr4i	⊢ ( 𝑉 ⊆ { 𝑥 ∈ 𝑉 ∣ 𝜑 } ↔ 𝑉 = { 𝑥 ∈ 𝑉 ∣ 𝜑 } )