Metamath Proof Explorer


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 ( ( 𝐴𝑉𝑅𝑊 ) → ≀ ( 𝑅 ⋉ ( E ↾ 𝐴 ) ) ∈ V )

Proof

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