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Theorem relres 4847
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 4551 . . 3 (𝐴𝐵) = (𝐴 ∩ (𝐵 × V))
2 inss2 3297 . . 3 (𝐴 ∩ (𝐵 × V)) ⊆ (𝐵 × V)
31, 2eqsstri 3129 . 2 (𝐴𝐵) ⊆ (𝐵 × V)
4 relxp 4648 . 2 Rel (𝐵 × V)
5 relss 4626 . 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 2686  cin 3070  wss 3071   × cxp 4537  cres 4541  Rel wrel 4544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-in 3077  df-ss 3084  df-opab 3990  df-xp 4545  df-rel 4546  df-res 4551
This theorem is referenced by:  elres  4855  resiexg  4864  iss  4865  dfres2  4871  issref  4921  asymref  4924  poirr2  4931  cnvcnvres  5002  resco  5043  ressn  5079  funssres  5165  fnresdisj  5233  fnres  5239  fcnvres  5306  nfunsn  5455  fsnunfv  5621  resfunexgALT  6008  setsresg  12011
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