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Theorem resabs1 4640
 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 4624 . 2 ((𝐴𝐶) ↾ 𝐵) = (𝐴 ↾ (𝐶𝐵))
2 sseqin2 3156 . . 3 (𝐵𝐶 ↔ (𝐶𝐵) = 𝐵)
3 reseq2 4607 . . 3 ((𝐶𝐵) = 𝐵 → (𝐴 ↾ (𝐶𝐵)) = (𝐴𝐵))
42, 3sylbi 114 . 2 (𝐵𝐶 → (𝐴 ↾ (𝐶𝐵)) = (𝐴𝐵))
51, 4syl5eq 2084 1 (𝐵𝐶 → ((𝐴𝐶) ↾ 𝐵) = (𝐴𝐵))
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1243   ∩ cin 2916   ⊆ wss 2917   ↾ cres 4347 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-opab 3819  df-xp 4351  df-rel 4352  df-res 4357 This theorem is referenced by:  resabs2  4641  resiima  4683  fun2ssres  4943  fssres2  5067  f2ndf  5847  smores3  5908  tfrlemisucaccv  5939
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