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Theorem opabiotadm 6308
Description: Define a function whose value is "the unique  y such that  ph ( x ,  y )". (Contributed by NM, 16-Nov-2013.)
Hypothesis
Ref Expression
opabiota.1  |-  F  =  { <. x ,  y
>.  |  { y  |  ph }  =  {
y } }
Assertion
Ref Expression
opabiotadm  |-  dom  F  =  { x  |  E! y ph }
Distinct variable group:    x, y, F
Allowed substitution hints:    ph( x, y)

Proof of Theorem opabiotadm
StepHypRef Expression
1 dmopab 4905 . 2  |-  dom  { <. x ,  y >.  |  { y  |  ph }  =  { y } }  =  {
x  |  E. y { y  |  ph }  =  { y } }
2 opabiota.1 . . 3  |-  F  =  { <. x ,  y
>.  |  { y  |  ph }  =  {
y } }
32dmeqi 4896 . 2  |-  dom  F  =  dom  { <. x ,  y >.  |  {
y  |  ph }  =  { y } }
4 euabsn 3712 . . 3  |-  ( E! y ph  <->  E. y { y  |  ph }  =  { y } )
54abbii 2408 . 2  |-  { x  |  E! y ph }  =  { x  |  E. y { y  |  ph }  =  { y } }
61, 3, 53eqtr4i 2326 1  |-  dom  F  =  { x  |  E! y ph }
Colors of variables: wff set class
Syntax hints:   E.wex 1531    = wceq 1632   E!weu 2156   {cab 2282   {csn 3653   {copab 4092   dom cdm 4705
This theorem is referenced by:  opabiota  6309
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-dm 4715
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