Theorem relres 6011

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 (𝐴 ↾ 𝐵)

Proof of Theorem relres

Step	Hyp	Ref	Expression
1		df-res 5689	. . 3 ⊢ (𝐴 ↾ 𝐵) = (𝐴 ∩ (𝐵 × V))
2		inss2 4230	. . 3 ⊢ (𝐴 ∩ (𝐵 × V)) ⊆ (𝐵 × V)
3	1, 2	eqsstri 4017	. 2 ⊢ (𝐴 ↾ 𝐵) ⊆ (𝐵 × V)
4		relxp 5695	. 2 ⊢ Rel (𝐵 × V)
5		relss 5782	. 2 ⊢ ((𝐴 ↾ 𝐵) ⊆ (𝐵 × V) → (Rel (𝐵 × V) → Rel (𝐴 ↾ 𝐵)))
6	3, 4, 5	mp2 9	1 ⊢ Rel (𝐴 ↾ 𝐵)

Syntax hints:  Vcvv 3475   ∩ cin 3948   ⊆ wss 3949   × cxp 5675   ↾ cres 5679  Rel wrel 5682

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-in 3956  df-ss 3966  df-opab 5212  df-xp 5683  df-rel 5684  df-res 5689

This theorem is referenced by:  relresdm1  6034  iss  6036  dfres2  6042  restidsing  6053  asymref  6118  poirr2  6126  cnvcnvres  6205  resco  6250  coeq0  6255  resssxp  6270  ressn  6285  dfpo2  6296  snres0  6298  funssres  6593  fnresdisj  6671  fnres  6678  fresaunres2  6764  fcnvres  6769  nfunsn  6934  dffv2  6987  fsnunfv  7185  eqfunresadj  7357  resfunexgALT  7934  domss2  9136  fidomdm  9329  ttrclco  9713  cottrcl  9714  dmttrcl  9716  rnttrcl  9717  frmin  9744  frrlem16  9753  frr1  9754  dmct  10519  relexp0rel  14984  setsres  17111  pospo  18298  metustid  24063  ovoliunlem1  25019  dvres  25428  dvres2  25429  dvlog  26159  efopnlem2  26165  noetasuplem2  27237  noetainflem2  27241  h2hlm  30233  hlimcaui  30489  dfrdg2  34767  funpartfun  34915  bj-idreseq  36043  bj-idreseqb  36044  brres2  37136  elecres  37145  br1cnvssrres  37375  refrelressn  37394  trrelressn  37453  dfeldisj2  37588  dfeldisj3  37589  dfeldisj4  37590  disjres  37614  antisymrelres  37633  antisymrelressn  37634  mapfzcons1  41455  diophrw  41497  eldioph2lem1  41498  eldioph2lem2  41499  undmrnresiss  42355  brfvrcld2  42443  relexpiidm  42455  limsupresuz  44419  liminfresuz  44500  funressnfv  45753  dfdfat2  45836  setrec2lem2  47739