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Theorem relres 4582
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 4300 . . 3 (AB) = (A ∩ (B × V))
2 inss2 3152 . . 3 (A ∩ (B × V)) ⊆ (B × V)
31, 2eqsstri 2969 . 2 (AB) ⊆ (B × V)
4 relxp 4390 . 2 Rel (B × V)
5 relss 4370 . 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 2551  cin 2910  wss 2911   × cxp 4286  cres 4290  Rel wrel 4293
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-in 2918  df-ss 2925  df-opab 3810  df-xp 4294  df-rel 4295  df-res 4300
This theorem is referenced by:  elres  4589  resiexg  4596  iss  4597  dfres2  4601  issref  4650  asymref  4653  poirr2  4660  cnvcnvres  4727  resco  4768  ressn  4801  funssres  4885  fnresdisj  4952  fnres  4958  fcnvres  5016  nfunsn  5150  fsnunfv  5306  resfunexgALT  5679
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