Theorem relopab 4739

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 2171	. 2 |- { <. x , y >. | ph } = { <. x , y >. | ph }
2	1	relopabi 4738	1 |- Rel { <. x , y >. | ph }

Syntax hints:   {copab 4050   Rel wrel 4617

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 705  ax-5 1441  ax-7 1442  ax-gen 1443  ax-ie1 1487  ax-ie2 1488  ax-8 1498  ax-10 1499  ax-11 1500  ax-i12 1501  ax-bndl 1503  ax-4 1504  ax-17 1520  ax-i9 1524  ax-ial 1528  ax-i5r 1529  ax-14 2145  ax-ext 2153  ax-sep 4108  ax-pow 4161  ax-pr 4195

This theorem depends on definitions:  df-bi 116  df-3an 976  df-tru 1352  df-nf 1455  df-sb 1757  df-clab 2158  df-cleq 2164  df-clel 2167  df-nfc 2302  df-ral 2454  df-rex 2455  df-v 2733  df-un 3126  df-in 3128  df-ss 3135  df-pw 3569  df-sn 3590  df-pr 3591  df-op 3593  df-opab 4052  df-xp 4618  df-rel 4619

This theorem is referenced by:  opabid2  4743  inopab  4744  difopab  4745  dfres2  4944  cnvopab  5014  funopab  5235  elopabi  6178  shftfn  10792  lmreltop  13072