Theorem nelrnres 42339

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 5872	. 2 ⊢ ran (𝐵 ↾ 𝐶) ⊆ ran 𝐵
2		ssnel 42202	. 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 2112   ⊆ wss 3853  ran crn 5537   ↾ cres 5538

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-ext 2708

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2073  df-clab 2715  df-cleq 2728  df-clel 2809  df-rab 3060  df-v 3400  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4224  df-if 4426  df-sn 4528  df-pr 4530  df-op 4534  df-br 5040  df-opab 5102  df-cnv 5544  df-dm 5546  df-rn 5547  df-res 5548

This theorem is referenced by:  sge0sup  43547  sge0less  43548  sge0resplit  43562