Theorem 1cossxrncnvepresex

Description: Sufficient condition for a restricted converse epsilon range Cartesian product to be a set. (Contributed by Peter Mazsa, 23-Sep-2021)

	Ref	Expression
Assertion	1cossxrncnvepresex	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊 ) → ≀ ( 𝑅 ⋉ ( ◡ E ↾ 𝐴 ) ) ∈ V )

Proof

Step	Hyp	Ref	Expression
1		xrncnvepresex	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊 ) → ( 𝑅 ⋉ ( ◡ E ↾ 𝐴 ) ) ∈ V )
2		cossex	⊢ ( ( 𝑅 ⋉ ( ◡ E ↾ 𝐴 ) ) ∈ V → ≀ ( 𝑅 ⋉ ( ◡ E ↾ 𝐴 ) ) ∈ V )
3	1 2	syl	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊 ) → ≀ ( 𝑅 ⋉ ( ◡ E ↾ 𝐴 ) ) ∈ V )