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Theorem relres 4667
Description: A restriction is a relation. Exercise 12 of [TakeutiZaring] p. 25. (Contributed by NM, 2-Aug-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
relres Rel (𝐴𝐵)

Proof of Theorem relres
StepHypRef Expression
1 df-res 4385 . . 3 (𝐴𝐵) = (𝐴 ∩ (𝐵 × V))
2 inss2 3186 . . 3 (𝐴 ∩ (𝐵 × V)) ⊆ (𝐵 × V)
31, 2eqsstri 3003 . 2 (𝐴𝐵) ⊆ (𝐵 × V)
4 relxp 4475 . 2 Rel (𝐵 × V)
5 relss 4455 . 2 ((𝐴𝐵) ⊆ (𝐵 × V) → (Rel (𝐵 × V) → Rel (𝐴𝐵)))
63, 4, 5mp2 16 1 Rel (𝐴𝐵)
Colors of variables: wff set class
Syntax hints:  Vcvv 2574  cin 2944  wss 2945   × cxp 4371  cres 4375  Rel wrel 4378
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-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-in 2952  df-ss 2959  df-opab 3847  df-xp 4379  df-rel 4380  df-res 4385
This theorem is referenced by:  elres  4674  resiexg  4681  iss  4682  dfres2  4686  issref  4735  asymref  4738  poirr2  4745  cnvcnvres  4812  resco  4853  ressn  4886  funssres  4970  fnresdisj  5037  fnres  5043  fcnvres  5101  nfunsn  5235  fsnunfv  5391  resfunexgALT  5765
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