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Theorem resabs2 5349
 Description: Absorption law for restriction. (Contributed by NM, 27-Mar-1998.)
Assertion
Ref Expression
resabs2 (𝐵𝐶 → ((𝐴𝐵) ↾ 𝐶) = (𝐴𝐵))

Proof of Theorem resabs2
StepHypRef Expression
1 rescom 5343 . 2 ((𝐴𝐵) ↾ 𝐶) = ((𝐴𝐶) ↾ 𝐵)
2 resabs1 5347 . 2 (𝐵𝐶 → ((𝐴𝐶) ↾ 𝐵) = (𝐴𝐵))
31, 2syl5eq 2656 1 (𝐵𝐶 → ((𝐴𝐵) ↾ 𝐶) = (𝐴𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ⊆ wss 3540   ↾ cres 5040 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-opab 4644  df-xp 5044  df-rel 5045  df-res 5050 This theorem is referenced by:  residm  5350  fresaunres2  5989  fourierdlem104  39103  fouriersw  39124
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