Theorem nelrnres 45933

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

Assertion

Ref	Expression
nelrnres	⊢ (¬ 𝐴 ∈ ran 𝐵 → ¬ 𝐴 ∈ ran (𝐵 ↾ 𝐶))

Proof of Theorem nelrnres

Step	Hyp	Ref	Expression
1		rnresss 6016	. 2 ⊢ ran (𝐵 ↾ 𝐶) ⊆ ran 𝐵
2		ssnel 45791	. 2 ⊢ ((ran (𝐵 ↾ 𝐶) ⊆ ran 𝐵 ∧ ¬ 𝐴 ∈ ran 𝐵) → ¬ 𝐴 ∈ ran (𝐵 ↾ 𝐶))
3	1, 2	mpan 702	1 ⊢ (¬ 𝐴 ∈ ran 𝐵 → ¬ 𝐴 ∈ ran (𝐵 ↾ 𝐶))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∈ wcel 2143   ⊆ wss 3905  ran crn 5662   ↾ cres 5663

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-br 5110  df-opab 5174  df-cnv 5669  df-dm 5671  df-rn 5672  df-res 5673

This theorem is used by:  sge0sup  47133  sge0less  47134  sge0resplit  47148