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Theorem resabs2d 42407
 Description: Absorption law for restriction. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypothesis
Ref Expression
resabs2d.1 (𝜑𝐵𝐶)
Assertion
Ref Expression
resabs2d (𝜑 → ((𝐴𝐵) ↾ 𝐶) = (𝐴𝐵))

Proof of Theorem resabs2d
StepHypRef Expression
1 resabs2d.1 . 2 (𝜑𝐵𝐶)
2 resabs2 5855 . 2 (𝐵𝐶 → ((𝐴𝐵) ↾ 𝐶) = (𝐴𝐵))
31, 2syl 17 1 (𝜑 → ((𝐴𝐵) ↾ 𝐶) = (𝐴𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ⊆ wss 3858   ↾ cres 5526 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 2729  ax-sep 5169  ax-nul 5176  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-rab 3079  df-v 3411  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-sn 4523  df-pr 4525  df-op 4529  df-opab 5095  df-xp 5530  df-rel 5531  df-res 5536 This theorem is referenced by: (None)
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