Theorem nelrnres 45639

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 5978	. 2 ⊢ ran (𝐵 ↾ 𝐶) ⊆ ran 𝐵
2		ssnel 45496	. 2 ⊢ ((ran (𝐵 ↾ 𝐶) ⊆ ran 𝐵 ∧ ¬ 𝐴 ∈ ran 𝐵) → ¬ 𝐴 ∈ ran (𝐵 ↾ 𝐶))
3	1, 2	mpan 691	1 ⊢ (¬ 𝐴 ∈ ran 𝐵 → ¬ 𝐴 ∈ ran (𝐵 ↾ 𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2114   ⊆ wss 3890  ran crn 5627   ↾ cres 5628

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-cnv 5634  df-dm 5636  df-rn 5637  df-res 5638

This theorem is referenced by:  sge0sup  46841  sge0less  46842  sge0resplit  46856