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