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Theorem oprab4 5719
Description: Two ways to state the domain of an operation. (Contributed by FL, 24-Jan-2010.)
Assertion
Ref Expression
oprab4  |-  { <. <.
x ,  y >. ,  z >.  |  (
<. x ,  y >.  e.  ( A  X.  B
)  /\  ph ) }  =  { <. <. x ,  y >. ,  z
>.  |  ( (
x  e.  A  /\  y  e.  B )  /\  ph ) }
Distinct variable group:    x, y, z
Allowed substitution hints:    ph( x, y, z)    A( x, y, z)    B( x, y, z)

Proof of Theorem oprab4
StepHypRef Expression
1 opelxp 4467 . . 3  |-  ( <.
x ,  y >.  e.  ( A  X.  B
)  <->  ( x  e.  A  /\  y  e.  B ) )
21anbi1i 446 . 2  |-  ( (
<. x ,  y >.  e.  ( A  X.  B
)  /\  ph )  <->  ( (
x  e.  A  /\  y  e.  B )  /\  ph ) )
32oprabbii 5704 1  |-  { <. <.
x ,  y >. ,  z >.  |  (
<. x ,  y >.  e.  ( A  X.  B
)  /\  ph ) }  =  { <. <. x ,  y >. ,  z
>.  |  ( (
x  e.  A  /\  y  e.  B )  /\  ph ) }
Colors of variables: wff set class
Syntax hints:    /\ wa 102    = wceq 1289    e. wcel 1438   <.cop 3449    X. cxp 4436   {coprab 5653
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-opab 3900  df-xp 4444  df-oprab 5656
This theorem is referenced by: (None)
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