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Theorem dmopab3 4548
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 2311 . 2  |-  ( A. x  e.  A  E. y ph  <->  A. x ( x  e.  A  ->  E. y ph ) )
2 pm4.71 369 . . 3  |-  ( ( x  e.  A  ->  E. y ph )  <->  ( x  e.  A  <->  ( x  e.  A  /\  E. y ph ) ) )
32albii 1359 . 2  |-  ( A. x ( x  e.  A  ->  E. y ph )  <->  A. x ( x  e.  A  <->  ( x  e.  A  /\  E. y ph ) ) )
4 dmopab 4546 . . . . 5  |-  dom  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) }  =  { x  |  E. y ( x  e.  A  /\  ph ) }
5 19.42v 1786 . . . . . 6  |-  ( E. y ( x  e.  A  /\  ph )  <->  ( x  e.  A  /\  E. y ph ) )
65abbii 2153 . . . . 5  |-  { x  |  E. y ( x  e.  A  /\  ph ) }  =  {
x  |  ( x  e.  A  /\  E. y ph ) }
74, 6eqtri 2060 . . . 4  |-  dom  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) }  =  { x  |  ( x  e.  A  /\  E. y ph ) }
87eqeq1i 2047 . . 3  |-  ( dom 
{ <. x ,  y
>.  |  ( x  e.  A  /\  ph ) }  =  A  <->  { x  |  ( x  e.  A  /\  E. y ph ) }  =  A )
9 eqcom 2042 . . 3  |-  ( A  =  { x  |  ( x  e.  A  /\  E. y ph ) } 
<->  { x  |  ( x  e.  A  /\  E. y ph ) }  =  A )
10 abeq2 2146 . . 3  |-  ( A  =  { x  |  ( x  e.  A  /\  E. y ph ) } 
<-> 
A. x ( x  e.  A  <->  ( x  e.  A  /\  E. y ph ) ) )
118, 9, 103bitr2ri 198 . 2  |-  ( A. x ( x  e.  A  <->  ( x  e.  A  /\  E. y ph ) )  <->  dom  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) }  =  A )
121, 3, 113bitri 195 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    /\ wa 97    <-> wb 98   A.wal 1241    = wceq 1243   E.wex 1381    e. wcel 1393   {cab 2026   A.wral 2306   {copab 3817   dom cdm 4345
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-dm 4355
This theorem is referenced by:  dmxpm  4555  fnopabg  5022
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