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Theorem resabs2 5398
 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 5392 . 2 ((𝐴𝐵) ↾ 𝐶) = ((𝐴𝐶) ↾ 𝐵)
2 resabs1 5396 . 2 (𝐵𝐶 → ((𝐴𝐶) ↾ 𝐵) = (𝐴𝐵))
31, 2syl5eq 2667 1 (𝐵𝐶 → ((𝐴𝐵) ↾ 𝐶) = (𝐴𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1480   ⊆ wss 3560   ↾ cres 5086 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-opab 4684  df-xp 5090  df-rel 5091  df-res 5096 This theorem is referenced by:  residm  5399  fresaunres2  6043  resabs2d  39128  fourierdlem104  39764  fouriersw  39785
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