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Theorem nelrnres 41809
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 5854 . 2 ran (𝐵𝐶) ⊆ ran 𝐵
2 ssnel 41669 . 2 ((ran (𝐵𝐶) ⊆ ran 𝐵 ∧ ¬ 𝐴 ∈ ran 𝐵) → ¬ 𝐴 ∈ ran (𝐵𝐶))
31, 2mpan 689 1 𝐴 ∈ ran 𝐵 → ¬ 𝐴 ∈ ran (𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wcel 2111  wss 3881  ran crn 5520  cres 5521
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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-v 3443  df-un 3886  df-in 3888  df-ss 3898  df-sn 4526  df-pr 4528  df-op 4532  df-br 5031  df-opab 5093  df-cnv 5527  df-dm 5529  df-rn 5530  df-res 5531
This theorem is referenced by:  sge0sup  43025  sge0less  43026  sge0resplit  43040
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