Theorem relopab 4904

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 2238	. 2 |- { <. x , y >. | ph } = { <. x , y >. | ph }
2	1	relopabi 4903	1 |- Rel { <. x , y >. | ph }

Syntax hints:   {copab 4189   Rel wrel 4777

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-opab 4191  df-xp 4778  df-rel 4779

This theorem is referenced by:  brabv  4905  opabid2  4909  inopab  4910  difopab  4911  dfres2  5113  cnvopab  5187  funopab  5410  relmptopab  6284  elopabi  6424  exmidapne  7619  shftfn  11570  releqgg  14003