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Theorem rescom 4663
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 3156 . . 3 (𝐵𝐶) = (𝐶𝐵)
21reseq2i 4636 . 2 (𝐴 ↾ (𝐵𝐶)) = (𝐴 ↾ (𝐶𝐵))
3 resres 4651 . 2 ((𝐴𝐵) ↾ 𝐶) = (𝐴 ↾ (𝐵𝐶))
4 resres 4651 . 2 ((𝐴𝐶) ↾ 𝐵) = (𝐴 ↾ (𝐶𝐵))
52, 3, 43eqtr4i 2086 1 ((𝐴𝐵) ↾ 𝐶) = ((𝐴𝐶) ↾ 𝐵)
Colors of variables: wff set class
Syntax hints:   = wceq 1259  cin 2943  cres 4374
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-opab 3846  df-xp 4378  df-rel 4379  df-res 4384
This theorem is referenced by:  resabs2  4668
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