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Theorem oprab4 6110
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 4875 . . 3  |-  ( <.
x ,  y >.  e.  ( A  X.  B
)  <->  ( x  e.  A  /\  y  e.  B ) )
21anbi1i 677 . 2  |-  ( (
<. x ,  y >.  e.  ( A  X.  B
)  /\  ph )  <->  ( (
x  e.  A  /\  y  e.  B )  /\  ph ) )
32oprabbii 6096 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 359    = wceq 1649    e. wcel 1721   <.cop 3785    X. cxp 4843   {coprab 6049
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pr 4371
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-opab 4235  df-xp 4851  df-oprab 6052
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