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Theorem oprab4 6146
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 4911 . . 3  |-  ( <.
x ,  y >.  e.  ( A  X.  B
)  <->  ( x  e.  A  /\  y  e.  B ) )
21anbi1i 678 . 2  |-  ( (
<. x ,  y >.  e.  ( A  X.  B
)  /\  ph )  <->  ( (
x  e.  A  /\  y  e.  B )  /\  ph ) )
32oprabbii 6132 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 360    = wceq 1653    e. wcel 1726   <.cop 3819    X. cxp 4879   {coprab 6085
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-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  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-opab 4270  df-xp 4887  df-oprab 6088
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