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Theorem dmopab3 5074
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 2702 . 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 1575 . 2  |-  ( A. x ( x  e.  A  ->  E. y ph )  <->  A. x ( x  e.  A  <->  ( x  e.  A  /\  E. y ph ) ) )
4 dmopab 5072 . . . . 5  |-  dom  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) }  =  { x  |  E. y ( x  e.  A  /\  ph ) }
5 19.42v 1928 . . . . . 6  |-  ( E. y ( x  e.  A  /\  ph )  <->  ( x  e.  A  /\  E. y ph ) )
65abbii 2547 . . . . 5  |-  { x  |  E. y ( x  e.  A  /\  ph ) }  =  {
x  |  ( x  e.  A  /\  E. y ph ) }
74, 6eqtri 2455 . . . 4  |-  dom  { <. x ,  y >.  |  ( x  e.  A  /\  ph ) }  =  { x  |  ( x  e.  A  /\  E. y ph ) }
87eqeq1i 2442 . . 3  |-  ( dom 
{ <. x ,  y
>.  |  ( x  e.  A  /\  ph ) }  =  A  <->  { x  |  ( x  e.  A  /\  E. y ph ) }  =  A )
9 eqcom 2437 . . 3  |-  ( A  =  { x  |  ( x  e.  A  /\  E. y ph ) } 
<->  { x  |  ( x  e.  A  /\  E. y ph ) }  =  A )
10 abeq2 2540 . . 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 1549   E.wex 1550    = wceq 1652    e. wcel 1725   {cab 2421   A.wral 2697   {copab 4257   dom cdm 4870
This theorem is referenced by:  dmxp  5080  fnopabg  5560
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-dm 4880
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