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Theorem residm 5970
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 3966 . 2 𝐵𝐵
2 resabs2 5969 . 2 (𝐵𝐵 → ((𝐴𝐵) ↾ 𝐵) = (𝐴𝐵))
31, 2ax-mp 5 1 ((𝐴𝐵) ↾ 𝐵) = (𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1541  wss 3910  cres 5635
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-opab 5168  df-xp 5639  df-rel 5640  df-res 5645
This theorem is referenced by:  resima  5971  dffv2  6936  fvsnun2  7128  qtopres  23047  bnj1253  33620  eldioph2lem1  41061  eldioph2lem2  41062  relexpiidm  41958
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