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Theorem dmopab3 4576
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 2328 . 2  |-  ( A. x  e.  A  E. y ph  <->  A. x ( x  e.  A  ->  E. y ph ) )
2 pm4.71 375 . . 3  |-  ( ( x  e.  A  ->  E. y ph )  <->  ( x  e.  A  <->  ( x  e.  A  /\  E. y ph ) ) )
32albii 1375 . 2  |-  ( A. x ( x  e.  A  ->  E. y ph )  <->  A. x ( x  e.  A  <->  ( x  e.  A  /\  E. y ph ) ) )
4 dmopab 4574 . . . . 5  |-  dom  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) }  =  { x  |  E. y ( x  e.  A  /\  ph ) }
5 19.42v 1802 . . . . . 6  |-  ( E. y ( x  e.  A  /\  ph )  <->  ( x  e.  A  /\  E. y ph ) )
65abbii 2169 . . . . 5  |-  { x  |  E. y ( x  e.  A  /\  ph ) }  =  {
x  |  ( x  e.  A  /\  E. y ph ) }
74, 6eqtri 2076 . . . 4  |-  dom  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) }  =  { x  |  ( x  e.  A  /\  E. y ph ) }
87eqeq1i 2063 . . 3  |-  ( dom 
{ <. x ,  y
>.  |  ( x  e.  A  /\  ph ) }  =  A  <->  { x  |  ( x  e.  A  /\  E. y ph ) }  =  A )
9 eqcom 2058 . . 3  |-  ( A  =  { x  |  ( x  e.  A  /\  E. y ph ) } 
<->  { x  |  ( x  e.  A  /\  E. y ph ) }  =  A )
10 abeq2 2162 . . 3  |-  ( A  =  { x  |  ( x  e.  A  /\  E. y ph ) } 
<-> 
A. x ( x  e.  A  <->  ( x  e.  A  /\  E. y ph ) ) )
118, 9, 103bitr2ri 202 . 2  |-  ( A. x ( x  e.  A  <->  ( x  e.  A  /\  E. y ph ) )  <->  dom  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) }  =  A )
121, 3, 113bitri 199 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 101    <-> wb 102   A.wal 1257    = wceq 1259   E.wex 1397    e. wcel 1409   {cab 2042   A.wral 2323   {copab 3845   dom cdm 4373
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-opab 3847  df-dm 4383
This theorem is referenced by:  dmxpm  4583  fnopabg  5050
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