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Theorem dmopab3 4854
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 2472 . 2  |-  ( A. x  e.  A  E. y ph  <->  A. x ( x  e.  A  ->  E. y ph ) )
2 pm4.71 389 . . 3  |-  ( ( x  e.  A  ->  E. y ph )  <->  ( x  e.  A  <->  ( x  e.  A  /\  E. y ph ) ) )
32albii 1480 . 2  |-  ( A. x ( x  e.  A  ->  E. y ph )  <->  A. x ( x  e.  A  <->  ( x  e.  A  /\  E. y ph ) ) )
4 dmopab 4852 . . . . 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 2304 . . . . 5  |-  { x  |  E. y ( x  e.  A  /\  ph ) }  =  {
x  |  ( x  e.  A  /\  E. y ph ) }
74, 6eqtri 2209 . . . 4  |-  dom  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) }  =  { x  |  ( x  e.  A  /\  E. y ph ) }
87eqeq1i 2196 . . 3  |-  ( dom 
{ <. x ,  y
>.  |  ( x  e.  A  /\  ph ) }  =  A  <->  { x  |  ( x  e.  A  /\  E. y ph ) }  =  A )
9 eqcom 2190 . . 3  |-  ( A  =  { x  |  ( x  e.  A  /\  E. y ph ) } 
<->  { x  |  ( x  e.  A  /\  E. y ph ) }  =  A )
10 abeq2 2297 . . 3  |-  ( A  =  { x  |  ( x  e.  A  /\  E. y ph ) } 
<-> 
A. x ( x  e.  A  <->  ( x  e.  A  /\  E. y ph ) ) )
118, 9, 103bitr2ri 209 . 2  |-  ( A. x ( x  e.  A  <->  ( x  e.  A  /\  E. y ph ) )  <->  dom  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) }  =  A )
121, 3, 113bitri 206 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 104    <-> wb 105   A.wal 1361    = wceq 1363   E.wex 1502    e. wcel 2159   {cab 2174   A.wral 2467   {copab 4077   dom cdm 4640
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2162  ax-ext 2170  ax-sep 4135  ax-pow 4188  ax-pr 4223
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2040  df-mo 2041  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-ral 2472  df-v 2753  df-un 3147  df-in 3149  df-ss 3156  df-pw 3591  df-sn 3612  df-pr 3613  df-op 3615  df-br 4018  df-opab 4079  df-dm 4650
This theorem is referenced by:  dmxpm  4861  dmxpid  4862  fnopabg  5353  acfun  7223  ccfunen  7280
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