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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 ⊢ A ∈ V ∧ R ∈ W → ≀ R ⋉ E -1 ↾ A ∈ V

Proof

Step Hyp Ref Expression
1 xrncnvepresex ⊢ A ∈ V ∧ R ∈ W → R ⋉ E -1 ↾ A ∈ V
2 cossex ⊢ R ⋉ E -1 ↾ A ∈ V → ≀ R ⋉ E -1 ↾ A ∈ V
3 1 2 syl ⊢ A ∈ V ∧ R ∈ W → ≀ R ⋉ E -1 ↾ A ∈ V