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Theorem resabs2d 41113
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 5730 . 2 (𝐵𝐶 → ((𝐴𝐵) ↾ 𝐶) = (𝐴𝐵))
31, 2syl 17 1 (𝜑 → ((𝐴𝐵) ↾ 𝐶) = (𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1507  wss 3829  cres 5409
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-ext 2750  ax-sep 5060  ax-nul 5067  ax-pr 5186
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-rab 3097  df-v 3417  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-nul 4179  df-if 4351  df-sn 4442  df-pr 4444  df-op 4448  df-opab 4992  df-xp 5413  df-rel 5414  df-res 5419
This theorem is referenced by: (None)
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