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Theorem dmoprabss 6365
Description: The domain of an operation class abstraction. (Contributed by NM, 24-Aug-1995.)
Assertion
Ref Expression
dmoprabss  |-  dom  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  A  /\  y  e.  B )  /\  ph ) }  C_  ( A  X.  B )
Distinct variable groups:    x, y,
z, A    x, B, y, z
Allowed substitution hints:    ph( x, y, z)

Proof of Theorem dmoprabss
StepHypRef Expression
1 dmoprab 6364 . 2  |-  dom  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  A  /\  y  e.  B )  /\  ph ) }  =  { <. x ,  y >.  |  E. z ( ( x  e.  A  /\  y  e.  B )  /\  ph ) }
2 19.42v 1799 . . . 4  |-  ( E. z ( ( x  e.  A  /\  y  e.  B )  /\  ph ) 
<->  ( ( x  e.  A  /\  y  e.  B )  /\  E. z ph ) )
32opabbii 4459 . . 3  |-  { <. x ,  y >.  |  E. z ( ( x  e.  A  /\  y  e.  B )  /\  ph ) }  =  { <. x ,  y >.  |  ( ( x  e.  A  /\  y  e.  B )  /\  E. z ph ) }
4 opabssxp 4898 . . 3  |-  { <. x ,  y >.  |  ( ( x  e.  A  /\  y  e.  B
)  /\  E. z ph ) }  C_  ( A  X.  B )
53, 4eqsstri 3472 . 2  |-  { <. x ,  y >.  |  E. z ( ( x  e.  A  /\  y  e.  B )  /\  ph ) }  C_  ( A  X.  B )
61, 5eqsstri 3472 1  |-  dom  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  A  /\  y  e.  B )  /\  ph ) }  C_  ( A  X.  B )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 367   E.wex 1633    e. wcel 1842    C_ wss 3414   {copab 4452    X. cxp 4821   dom cdm 4823   {coprab 6279
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pr 4630
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-rab 2763  df-v 3061  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-br 4396  df-opab 4454  df-xp 4829  df-dm 4833  df-oprab 6282
This theorem is referenced by:  mpt2ndm0  6497  elmpt2cl  6498  oprabexd  6771  oprabex  6772  bropopvvv  6864  dmaddsr  9492  dmmulsr  9493  axaddf  9552  axmulf  9553  2wlkonot3v  25292  2spthonot3v  25293
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