Theorem nelrnres 44608

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 6026	. 2 ⊢ ran (𝐵 ↾ 𝐶) ⊆ ran 𝐵
2		ssnel 44454	. 2 ⊢ ((ran (𝐵 ↾ 𝐶) ⊆ ran 𝐵 ∧ ¬ 𝐴 ∈ ran 𝐵) → ¬ 𝐴 ∈ ran (𝐵 ↾ 𝐶))
3	1, 2	mpan 688	1 ⊢ (¬ 𝐴 ∈ ran 𝐵 → ¬ 𝐴 ∈ ran (𝐵 ↾ 𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2098   ⊆ wss 3949  ran crn 5683   ↾ cres 5684

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-br 5153  df-opab 5215  df-cnv 5690  df-dm 5692  df-rn 5693  df-res 5694

This theorem is referenced by:  sge0sup  45826  sge0less  45827  sge0resplit  45841