Theorem reloprab 6068

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 6067	. 2 |- { <. <. x , y >. , z >. | ph } = { <. w , z >. | E. x E. y ( w = <. x , y >. /\ ph ) }
2	1	relopabi 4855	1 |- Rel { <. <. x , y >. , z >. | ph }

Syntax hints:   /\ wa 104   = wceq 1397   E.wex 1540   <.cop 3672   Rel wrel 4730   {coprab 6018

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-opab 4151  df-xp 4731  df-rel 4732  df-oprab 6021

This theorem is referenced by:  oprabss  6106