Theorem relopab 5660

Description: A class of ordered pairs is a relation. (Contributed by NM, 8-Mar-1995.) Remove disjoint variable conditions. (Revised by Alan Sare, 9-Jul-2013.) (Proof shortened by Mario Carneiro, 21-Dec-2013.)

Assertion

Ref	Expression
relopab	⊢ Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑}

Proof of Theorem relopab

Step	Hyp	Ref	Expression
1		eqid 2798	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑥, 𝑦⟩ ∣ 𝜑}
2	1	relopabi 5658	1 ⊢ Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑}

Syntax hints:  {copab 5092  Rel wrel 5524

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-11 2158  ax-12 2175  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3443  df-un 3886  df-in 3888  df-ss 3898  df-sn 4526  df-pr 4528  df-op 4532  df-opab 5093  df-xp 5525  df-rel 5526

This theorem is referenced by:  opabid2  5664  inopab  5665  difopab  5666  dfres2  5876  cnvopab  5964  funopab  6359  relmptopab  7375  elopabi  7742  relmpoopab  7772  shftfn  14424  cicer  17068  joindmss  17609  meetdmss  17623  lgsquadlem3  25966  tgjustf  26267  perpln1  26504  perpln2  26505  fpwrelmapffslem  30494  fpwrelmap  30495  relfae  31616  satfrel  32727  bj-0nelopab  34482  vvdifopab  35681  inxprnres  35709  prtlem12  36163  dicvalrelN  38481  diclspsn  38490  dih1dimatlem  38625  rfovcnvf1od  40705