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Theorem resabs1 4742
Description: Absorption law for restriction. Exercise 17 of [TakeutiZaring] p. 25. (Contributed by NM, 9-Aug-1994.)
Assertion
Ref Expression
resabs1 (𝐵𝐶 → ((𝐴𝐶) ↾ 𝐵) = (𝐴𝐵))

Proof of Theorem resabs1
StepHypRef Expression
1 resres 4725 . 2 ((𝐴𝐶) ↾ 𝐵) = (𝐴 ↾ (𝐶𝐵))
2 sseqin2 3219 . . 3 (𝐵𝐶 ↔ (𝐶𝐵) = 𝐵)
3 reseq2 4708 . . 3 ((𝐶𝐵) = 𝐵 → (𝐴 ↾ (𝐶𝐵)) = (𝐴𝐵))
42, 3sylbi 119 . 2 (𝐵𝐶 → (𝐴 ↾ (𝐶𝐵)) = (𝐴𝐵))
51, 4syl5eq 2132 1 (𝐵𝐶 → ((𝐴𝐶) ↾ 𝐵) = (𝐴𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1289  cin 2998  wss 2999  cres 4440
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-opab 3900  df-xp 4444  df-rel 4445  df-res 4450
This theorem is referenced by:  resabs2  4743  resiima  4790  fun2ssres  5057  fssres2  5188  f2ndf  5991  smores3  6058  tfrlemisucaccv  6090  tfr1onlemsucaccv  6106  tfrcllemsucaccv  6119  djudom  6785  setsresg  11527
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