Description: A completely normal space is normal. (Contributed by Mario Carneiro, 26-Aug-2015)
Ref | Expression | ||
---|---|---|---|
Assertion | cnrmnrm | ⊢ ( 𝐽 ∈ CNrm → 𝐽 ∈ Nrm ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid | ⊢ ∪ 𝐽 = ∪ 𝐽 | |
2 | 1 | restid | ⊢ ( 𝐽 ∈ CNrm → ( 𝐽 ↾t ∪ 𝐽 ) = 𝐽 ) |
3 | uniexg | ⊢ ( 𝐽 ∈ CNrm → ∪ 𝐽 ∈ V ) | |
4 | cnrmi | ⊢ ( ( 𝐽 ∈ CNrm ∧ ∪ 𝐽 ∈ V ) → ( 𝐽 ↾t ∪ 𝐽 ) ∈ Nrm ) | |
5 | 3 4 | mpdan | ⊢ ( 𝐽 ∈ CNrm → ( 𝐽 ↾t ∪ 𝐽 ) ∈ Nrm ) |
6 | 2 5 | eqeltrrd | ⊢ ( 𝐽 ∈ CNrm → 𝐽 ∈ Nrm ) |