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Theorem rescom 5848
Description: Commutative law for restriction. (Contributed by NM, 27-Mar-1998.)
Assertion
Ref Expression
rescom ((𝐴𝐵) ↾ 𝐶) = ((𝐴𝐶) ↾ 𝐵)

Proof of Theorem rescom
StepHypRef Expression
1 incom 4131 . . 3 (𝐵𝐶) = (𝐶𝐵)
21reseq2i 5819 . 2 (𝐴 ↾ (𝐵𝐶)) = (𝐴 ↾ (𝐶𝐵))
3 resres 5835 . 2 ((𝐴𝐵) ↾ 𝐶) = (𝐴 ↾ (𝐵𝐶))
4 resres 5835 . 2 ((𝐴𝐶) ↾ 𝐵) = (𝐴 ↾ (𝐶𝐵))
52, 3, 43eqtr4i 2834 1 ((𝐴𝐵) ↾ 𝐶) = ((𝐴𝐶) ↾ 𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1538  cin 3883  cres 5525
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-rab 3118  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-opab 5096  df-xp 5529  df-rel 5530  df-res 5535
This theorem is referenced by:  resabs2  5854  setscom  16523  dvres3a  24521  cpnres  24544  dvmptres3  24563  limsupresuz  42342  liminfresuz  42423
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