MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  residm Structured version   Visualization version   GIF version

Theorem residm 6030
Description: Idempotent law for restriction. (Contributed by NM, 27-Mar-1998.)
Assertion
Ref Expression
residm ((𝐴𝐵) ↾ 𝐵) = (𝐴𝐵)

Proof of Theorem residm
StepHypRef Expression
1 ssid 4018 . 2 𝐵𝐵
2 resabs2 6029 . 2 (𝐵𝐵 → ((𝐴𝐵) ↾ 𝐵) = (𝐴𝐵))
31, 2ax-mp 5 1 ((𝐴𝐵) ↾ 𝐵) = (𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  wss 3963  cres 5691
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-opab 5211  df-xp 5695  df-rel 5696  df-res 5701
This theorem is referenced by:  resima  6035  dffv2  7004  fvsnun2  7203  qtopres  23722  bnj1253  35010  eldioph2lem1  42748  eldioph2lem2  42749  relexpiidm  43694
  Copyright terms: Public domain W3C validator