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Theorem rescom 4684
 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 3174 . . 3 (𝐵𝐶) = (𝐶𝐵)
21reseq2i 4657 . 2 (𝐴 ↾ (𝐵𝐶)) = (𝐴 ↾ (𝐶𝐵))
3 resres 4672 . 2 ((𝐴𝐵) ↾ 𝐶) = (𝐴 ↾ (𝐵𝐶))
4 resres 4672 . 2 ((𝐴𝐶) ↾ 𝐵) = (𝐴 ↾ (𝐶𝐵))
52, 3, 43eqtr4i 2113 1 ((𝐴𝐵) ↾ 𝐶) = ((𝐴𝐶) ↾ 𝐵)
 Colors of variables: wff set class Syntax hints:   = wceq 1285   ∩ cin 2981   ↾ cres 4393 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992 This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-opab 3860  df-xp 4397  df-rel 4398  df-res 4403 This theorem is referenced by:  resabs2  4689
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