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Theorem dmoprabss 6283
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 6282 . 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 1936 . . . 4  |-  ( E. z ( ( x  e.  A  /\  y  e.  B )  /\  ph ) 
<->  ( ( x  e.  A  /\  y  e.  B )  /\  E. z ph ) )
32opabbii 4465 . . 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 5020 . . 3  |-  { <. x ,  y >.  |  ( ( x  e.  A  /\  y  e.  B
)  /\  E. z ph ) }  C_  ( A  X.  B )
53, 4eqsstri 3495 . 2  |-  { <. x ,  y >.  |  E. z ( ( x  e.  A  /\  y  e.  B )  /\  ph ) }  C_  ( A  X.  B )
61, 5eqsstri 3495 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 369   E.wex 1587    e. wcel 1758    C_ wss 3437   {copab 4458    X. cxp 4947   dom cdm 4949   {coprab 6202
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pr 4640
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-rab 2808  df-v 3080  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-op 3993  df-br 4402  df-opab 4460  df-xp 4955  df-dm 4959  df-oprab 6205
This theorem is referenced by:  elmpt2cl  6415  oprabexd  6675  oprabex  6676  bropopvvv  6764  mpt2ndm0  6850  dmaddsr  9364  dmmulsr  9365  axaddf  9424  axmulf  9425  2wlkonot3v  30543  2spthonot3v  30544
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