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Metamath Proof Explorer

Theorem nelrnres

Description: If A is not in the range, it is not in the range of any restriction. (Contributed by Glauco Siliprandi, 17-Aug-2020)

Ref Expression
Assertion nelrnres ⊢ ¬ A ∈ ran ⁡ B → ¬ A ∈ ran ⁡ B ↾ C

Proof

Step Hyp Ref Expression
1 rnresss ⊢ ran ⁡ B ↾ C ⊆ ran ⁡ B
2 ssnel ⊢ ran ⁡ B ↾ C ⊆ ran ⁡ B ∧ ¬ A ∈ ran ⁡ B → ¬ A ∈ ran ⁡ B ↾ C
3 1 2 mpan ⊢ ¬ A ∈ ran ⁡ B → ¬ A ∈ ran ⁡ B ↾ C