Theorem nelrnres 45191

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 6009	. 2 ⊢ ran (𝐵 ↾ 𝐶) ⊆ ran 𝐵
2		ssnel 45047	. 2 ⊢ ((ran (𝐵 ↾ 𝐶) ⊆ ran 𝐵 ∧ ¬ 𝐴 ∈ ran 𝐵) → ¬ 𝐴 ∈ ran (𝐵 ↾ 𝐶))
3	1, 2	mpan 690	1 ⊢ (¬ 𝐴 ∈ ran 𝐵 → ¬ 𝐴 ∈ ran (𝐵 ↾ 𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2109   ⊆ wss 3931  ran crn 5660   ↾ cres 5661

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-br 5125  df-opab 5187  df-cnv 5667  df-dm 5669  df-rn 5670  df-res 5671

This theorem is referenced by:  sge0sup  46400  sge0less  46401  sge0resplit  46415