Theorem reloprab 6079

Description: An operation class abstraction is a relation. (Contributed by NM, 16-Jun-2004.)

Assertion

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

Distinct variable groups:   x, z   y, z

Allowed substitution hints:   ph( x, y, z)

Proof of Theorem reloprab

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfoprab2 6078	. 2 |- { <. <. x , y >. , z >. | ph } = { <. w , z >. | E. x E. y ( w = <. x , y >. /\ ph ) }
2	1	relopabi 4861	1 |- Rel { <. <. x , y >. , z >. | ph }

Syntax hints:   /\ wa 104   = wceq 1398   E.wex 1541   <.cop 3676   Rel wrel 4736   {coprab 6029

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-opab 4156  df-xp 4737  df-rel 4738  df-oprab 6032

This theorem is referenced by:  oprabss  6117