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Theorem nelrnres 38369
 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 (𝐵𝐶))

Proof of Theorem nelrnres
StepHypRef Expression
1 rnresss 38360 . . 3 ran (𝐵𝐶) ⊆ ran 𝐵
21a1i 11 . 2 𝐴 ∈ ran 𝐵 → ran (𝐵𝐶) ⊆ ran 𝐵)
3 id 22 . 2 𝐴 ∈ ran 𝐵 → ¬ 𝐴 ∈ ran 𝐵)
4 ssnel 38227 . 2 ((ran (𝐵𝐶) ⊆ ran 𝐵 ∧ ¬ 𝐴 ∈ ran 𝐵) → ¬ 𝐴 ∈ ran (𝐵𝐶))
52, 3, 4syl2anc 691 1 𝐴 ∈ ran 𝐵 → ¬ 𝐴 ∈ ran (𝐵𝐶))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 1977   ⊆ wss 3540  ran crn 5039   ↾ cres 5040 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-cnv 5046  df-dm 5048  df-rn 5049  df-res 5050 This theorem is referenced by:  sge0sup  39284  sge0less  39285  sge0resplit  39299
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