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Theorem dmopab3 5023
Description: The domain of a restricted class of ordered pairs. (Contributed by NM, 31-Jan-2004.)
Assertion
Ref Expression
dmopab3  |-  ( A. x  e.  A  E. y ph  <->  dom  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) }  =  A )
Distinct variable group:    x, y, A
Allowed substitution hints:    ph( x, y)

Proof of Theorem dmopab3
StepHypRef Expression
1 df-ral 2655 . 2  |-  ( A. x  e.  A  E. y ph  <->  A. x ( x  e.  A  ->  E. y ph ) )
2 pm4.71 612 . . 3  |-  ( ( x  e.  A  ->  E. y ph )  <->  ( x  e.  A  <->  ( x  e.  A  /\  E. y ph ) ) )
32albii 1572 . 2  |-  ( A. x ( x  e.  A  ->  E. y ph )  <->  A. x ( x  e.  A  <->  ( x  e.  A  /\  E. y ph ) ) )
4 dmopab 5021 . . . . 5  |-  dom  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) }  =  { x  |  E. y ( x  e.  A  /\  ph ) }
5 19.42v 1917 . . . . . 6  |-  ( E. y ( x  e.  A  /\  ph )  <->  ( x  e.  A  /\  E. y ph ) )
65abbii 2500 . . . . 5  |-  { x  |  E. y ( x  e.  A  /\  ph ) }  =  {
x  |  ( x  e.  A  /\  E. y ph ) }
74, 6eqtri 2408 . . . 4  |-  dom  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) }  =  { x  |  ( x  e.  A  /\  E. y ph ) }
87eqeq1i 2395 . . 3  |-  ( dom 
{ <. x ,  y
>.  |  ( x  e.  A  /\  ph ) }  =  A  <->  { x  |  ( x  e.  A  /\  E. y ph ) }  =  A )
9 eqcom 2390 . . 3  |-  ( A  =  { x  |  ( x  e.  A  /\  E. y ph ) } 
<->  { x  |  ( x  e.  A  /\  E. y ph ) }  =  A )
10 abeq2 2493 . . 3  |-  ( A  =  { x  |  ( x  e.  A  /\  E. y ph ) } 
<-> 
A. x ( x  e.  A  <->  ( x  e.  A  /\  E. y ph ) ) )
118, 9, 103bitr2ri 266 . 2  |-  ( A. x ( x  e.  A  <->  ( x  e.  A  /\  E. y ph ) )  <->  dom  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) }  =  A )
121, 3, 113bitri 263 1  |-  ( A. x  e.  A  E. y ph  <->  dom  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) }  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   A.wal 1546   E.wex 1547    = wceq 1649    e. wcel 1717   {cab 2374   A.wral 2650   {copab 4207   dom cdm 4819
This theorem is referenced by:  dmxp  5029  fnopabg  5509
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 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pr 4345
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-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rab 2659  df-v 2902  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-br 4155  df-opab 4209  df-dm 4829
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