Metamath Proof Explorer


Theorem xrncnvepresex

Description: Sufficient condition for a range Cartesian product with restricted converse epsilon to be a set. (Contributed by Peter Mazsa, 16-Dec-2020) (Revised by Peter Mazsa, 23-Sep-2021)

Ref Expression
Assertion xrncnvepresex ( ( 𝐴𝑉𝑅𝑊 ) → ( 𝑅 ⋉ ( E ↾ 𝐴 ) ) ∈ V )

Proof

Step Hyp Ref Expression
1 cnvepresex ( 𝐴𝑉 → ( E ↾ 𝐴 ) ∈ V )
2 1 adantr ( ( 𝐴𝑉𝑅𝑊 ) → ( E ↾ 𝐴 ) ∈ V )
3 xrnresex ( ( 𝐴𝑉𝑅𝑊 ∧ ( E ↾ 𝐴 ) ∈ V ) → ( 𝑅 ⋉ ( E ↾ 𝐴 ) ) ∈ V )
4 2 3 mpd3an3 ( ( 𝐴𝑉𝑅𝑊 ) → ( 𝑅 ⋉ ( E ↾ 𝐴 ) ) ∈ V )