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	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝑅 ∈ 𝑊 ) → ( 𝑅 ⋉ ( ◡ E ↾ 𝐴 ) ) ∈ V )

Proof

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