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Theorem rescom 5597
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 3966 . . 3 (𝐵𝐶) = (𝐶𝐵)
21reseq2i 5561 . 2 (𝐴 ↾ (𝐵𝐶)) = (𝐴 ↾ (𝐶𝐵))
3 resres 5584 . 2 ((𝐴𝐵) ↾ 𝐶) = (𝐴 ↾ (𝐵𝐶))
4 resres 5584 . 2 ((𝐴𝐶) ↾ 𝐵) = (𝐴 ↾ (𝐶𝐵))
52, 3, 43eqtr4i 2796 1 ((𝐴𝐵) ↾ 𝐶) = ((𝐴𝐶) ↾ 𝐵)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1652  cin 3730  cres 5278
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2349  ax-ext 2742  ax-sep 4940  ax-nul 4948  ax-pr 5061
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-clab 2751  df-cleq 2757  df-clel 2760  df-nfc 2895  df-rab 3063  df-v 3351  df-dif 3734  df-un 3736  df-in 3738  df-ss 3745  df-nul 4079  df-if 4243  df-sn 4334  df-pr 4336  df-op 4340  df-opab 4871  df-xp 5282  df-rel 5283  df-res 5288
This theorem is referenced by:  resabs2  5603  setscom  16176  dvres3a  23968  cpnres  23990  dvmptres3  24009  limsupresuz  40505  liminfresuz  40586
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