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Theorem rescom 5411
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 3797 . . 3 (𝐵𝐶) = (𝐶𝐵)
21reseq2i 5382 . 2 (𝐴 ↾ (𝐵𝐶)) = (𝐴 ↾ (𝐶𝐵))
3 resres 5397 . 2 ((𝐴𝐵) ↾ 𝐶) = (𝐴 ↾ (𝐵𝐶))
4 resres 5397 . 2 ((𝐴𝐶) ↾ 𝐵) = (𝐴 ↾ (𝐶𝐵))
52, 3, 43eqtr4i 2652 1 ((𝐴𝐵) ↾ 𝐶) = ((𝐴𝐶) ↾ 𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1481  cin 3566  cres 5106
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-rab 2918  df-v 3197  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-opab 4704  df-xp 5110  df-rel 5111  df-res 5116
This theorem is referenced by:  resabs2  5417  setscom  15884  dvres3a  23659  cpnres  23681  dvmptres3  23700  limsupresuz  39735  liminfresuz  39810
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