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Metamath Proof Explorer

Theorem xrnidresex

Description: Sufficient condition for a range Cartesian product with restricted identity to be a set. (Contributed by Peter Mazsa, 31-Dec-2021)

Ref Expression
Assertion xrnidresex ⊢ A ∈ V ∧ R ∈ W → R ⋉ I ↾ A ∈ V

Proof

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