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Theorem dmoprabss 6147
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 6146 . 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 1928 . . . 4  |-  ( E. z ( ( x  e.  A  /\  y  e.  B )  /\  ph ) 
<->  ( ( x  e.  A  /\  y  e.  B )  /\  E. z ph ) )
32opabbii 4264 . . 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 4942 . . 3  |-  { <. x ,  y >.  |  ( ( x  e.  A  /\  y  e.  B
)  /\  E. z ph ) }  C_  ( A  X.  B )
53, 4eqsstri 3370 . 2  |-  { <. x ,  y >.  |  E. z ( ( x  e.  A  /\  y  e.  B )  /\  ph ) }  C_  ( A  X.  B )
61, 5eqsstri 3370 1  |-  dom  { <. <. x ,  y
>. ,  z >.  |  ( ( x  e.  A  /\  y  e.  B )  /\  ph ) }  C_  ( A  X.  B )
Colors of variables: wff set class
Syntax hints:    /\ wa 359   E.wex 1550    e. wcel 1725    C_ wss 3312   {copab 4257    X. cxp 4868   dom cdm 4870   {coprab 6074
This theorem is referenced by:  oprabexd  6178  oprabex  6179  elmpt2cl  6280  bropopvvv  6418  mpt2ndm0  6465  dmaddsr  8952  dmmulsr  8953  axaddf  9012  axmulf  9013  2wlkonot3v  28295  2spthonot3v  28296
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-xp 4876  df-dm 4880  df-oprab 6077
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