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Theorem dmoprabss 6158
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 6157 . 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 1929 . . . 4  |-  ( E. z ( ( x  e.  A  /\  y  e.  B )  /\  ph ) 
<->  ( ( x  e.  A  /\  y  e.  B )  /\  E. z ph ) )
32opabbii 4275 . . 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 4953 . . 3  |-  { <. x ,  y >.  |  ( ( x  e.  A  /\  y  e.  B
)  /\  E. z ph ) }  C_  ( A  X.  B )
53, 4eqsstri 3380 . 2  |-  { <. x ,  y >.  |  E. z ( ( x  e.  A  /\  y  e.  B )  /\  ph ) }  C_  ( A  X.  B )
61, 5eqsstri 3380 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 360   E.wex 1551    e. wcel 1726    C_ wss 3322   {copab 4268    X. cxp 4879   dom cdm 4881   {coprab 6085
This theorem is referenced by:  oprabexd  6189  oprabex  6190  elmpt2cl  6291  bropopvvv  6429  mpt2ndm0  6476  dmaddsr  8965  dmmulsr  8966  axaddf  9025  axmulf  9026  2wlkonot3v  28407  2spthonot3v  28408
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4216  df-opab 4270  df-xp 4887  df-dm 4891  df-oprab 6088
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