Theorem reloprab 3977

Description: An operation class abstraction is a relation.

Assertion

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

Distinct variable groups:   x,z   y,z

Proof of Theorem reloprab

Step	Hyp	Ref	Expression
1		relopab 3256	. 2 |- Rel {<.w, z>. | E.xE.y(w = <.x, y>. /\ ph)}
2		dfoprab2 3976	. . 3 |- {<.<.x, y>., z>. | ph} = {<.w, z>. | E.xE.y(w = <.x, y>. /\ ph)}
3	2	releqi 3234	. 2 |- (Rel {<.<.x, y>., z>. | ph} <-> Rel {<.w, z>. | E.xE.y(w = <.x, y>. /\ ph)})
4	1, 3	mpbir 190	1 |- Rel {<.<.x, y>., z>. | ph}

Syntax hints:   /\ wa 223   = wceq 953  E.wex 977  <.cop 2401  {copab 2656  Rel wrel 3165  {copab2 3949

This theorem is referenced by:  oprabss 3991  eloprabi 4102

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-opab 2657  df-xp 3174  df-rel 3175  df-oprab 3951

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