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 ⊢ A ∈ V ∧ R ∈ W → R ⋉ E -1 ↾ A ∈ V

Proof

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