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Theorem resabs1i 41733
Description: Absorption law for restriction. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
Hypothesis
Ref Expression
resabs1i.1 𝐵𝐶
Assertion
Ref Expression
resabs1i ((𝐴𝐶) ↾ 𝐵) = (𝐴𝐵)

Proof of Theorem resabs1i
StepHypRef Expression
1 resabs1i.1 . 2 𝐵𝐶
2 resabs1 5872 . 2 (𝐵𝐶 → ((𝐴𝐶) ↾ 𝐵) = (𝐴𝐵))
31, 2ax-mp 5 1 ((𝐴𝐶) ↾ 𝐵) = (𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1538  wss 3919  cres 5545
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pr 5318
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 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-rab 3142  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-opab 5116  df-xp 5549  df-rel 5550  df-res 5555
This theorem is referenced by:  liminfresre  42374
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