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Theorem nelrnres 40297
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 40288 . 2 ran (𝐵𝐶) ⊆ ran 𝐵
2 ssnel 40137 . 2 ((ran (𝐵𝐶) ⊆ ran 𝐵 ∧ ¬ 𝐴 ∈ ran 𝐵) → ¬ 𝐴 ∈ ran (𝐵𝐶))
31, 2mpan 680 1 𝐴 ∈ ran 𝐵 → ¬ 𝐴 ∈ ran (𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wcel 2107  wss 3792  ran crn 5356  cres 5357
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-rab 3099  df-v 3400  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-br 4887  df-opab 4949  df-cnv 5363  df-dm 5365  df-rn 5366  df-res 5367
This theorem is referenced by:  sge0sup  41532  sge0less  41533  sge0resplit  41547
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