Theorem relopab 4978

Description: A class of ordered pairs is a relation. (Contributed by NM, 8-Mar-1995.) (Unnecessary distinct variable restrictions were removed by Alan Sare, 9-Jul-2013.) (Proof shortened by Mario Carneiro, 21-Dec-2013.)

Assertion

Ref	Expression
relopab	|- Rel { <. x , y >. | ph }

Proof of Theorem relopab

Step	Hyp	Ref	Expression
1		eqid 2461	. 2 |- { <. x , y >. | ph } = { <. x , y >. | ph }
2	1	relopabi 4977	1 |- Rel { <. x , y >. | ph }

Syntax hints:   {copab 4473   Rel wrel 4857

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-sep 4538  ax-nul 4547  ax-pr 4652

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-ral 2753  df-rex 2754  df-rab 2757  df-v 3058  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-nul 3743  df-if 3893  df-sn 3980  df-pr 3982  df-op 3986  df-opab 4475  df-xp 4858  df-rel 4859

This theorem is referenced by:  opabid2  4982  inopab  4983  difopab  4984  dfres2  5175  cnvopab  5255  funopab  5633  relmptopab  6543  elopabi  6880  relmpt2opab  6904  shftfn  13184  cicer  15759  joindmss  16301  meetdmss  16315  lgsquadlem3  24332  perpln1  24803  perpln2  24804  fpwrelmapffslem  28365  fpwrelmap  28366  relfae  29118  prtlem12  32483  dicvalrelN  34797  diclspsn  34806  dih1dimatlem  34941  rel1wlk  39688