Theorem relres 6012

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.) (Proof shortened by BJ, 16-Oct-2026.)

Assertion

Ref	Expression
relres	⊢ Rel (𝐴 ↾ 𝐵)

Proof of Theorem relres

Step	Hyp	Ref	Expression
1		relinxp 5801	. 2 ⊢ Rel (𝐴 ∩ (𝐵 × V))
2		df-res 5670	. . 3 ⊢ (𝐴 ↾ 𝐵) = (𝐴 ∩ (𝐵 × V))
3	2	releqi 5762	. 2 ⊢ (Rel (𝐴 ↾ 𝐵) ↔ Rel (𝐴 ∩ (𝐵 × V)))
4	1, 3	mpbir 234	1 ⊢ Rel (𝐴 ↾ 𝐵)

This proof depends on syntax axioms:  Vcvv 3454   ∩ cin 4092   × cxp 5656   ↾ cres 5660  Rel wrel 5663

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-in 4094  df-opab 5174  df-xp 5664  df-rel 5665  df-res 5670

This theorem is used by:  resindm  6037  relresdm1  6043  iss  6045  dfres2  6051  restidsing  6063  asymref  6128  poirr2  6136  cnvcnvres  6225  resco  6270  coeq0  6276  resssxp  6296  ressn  6312  dfpo2  6325  snres0  6327  funssres  6612  fnresdisj  6689  fnres  6696  fresaunres2  6784  fcnvres  6789  nfunsn  6955  dffv2  7011  fsnunfv  7223  eqfunresadj  7401  resfunexgALT  7994  elecres  8804  domss2  9193  fidomdm  9361  ttrclco  9756  cottrcl  9757  dmttrcl  9759  rnttrcl  9760  frmin  9790  frrlem16  9799  frr1  9800  setrec2lem2  10013  dmct  10641  dmctOLD  10642  relexp0rel  15233  setsres  17400  pospo  18567  metustid  24928  ovoliunlem1  25878  dvres  26286  dvres2  26287  dvlog  27036  efopnlem2  27042  noetasuplem2  28148  noetainflem2  28152  h2hlm  31639  hlimcaui  31895  dfrdg2  36772  funpartfun  36922  cossidcnvss2  38222  nvo3  38230  bj-idreseq  38351  bj-idreseqb  38352  brres2  39471  br1cnvssrres  39783  refrelressn  39802  trrelressn  39865  dfeldisj2  40008  dfeldisj3  40009  dfeldisj4  40010  disjres  40042  antisymrelres  40064  antisymrelressn  40065  mapfzcons1  43994  diophrw  44036  eldioph2lem1  44037  eldioph2lem2  44038  undmrnresiss  44865  brfvrcld2  44953  relexpiidm  44965  limsupresuz  46986  liminfresuz  47067  funressnfv  48386  dfdfat2  48471  resinsn  50316  resinsnALT  50317  tposres0  50321