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Theorem opabiotadm 6474
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 5021 . 2  |-  dom  { <. x ,  y >.  |  { y  |  ph }  =  { y } }  =  {
x  |  E. y { y  |  ph }  =  { y } }
2 opabiota.1 . . 3  |-  F  =  { <. x ,  y
>.  |  { y  |  ph }  =  {
y } }
32dmeqi 5012 . 2  |-  dom  F  =  dom  { <. x ,  y >.  |  {
y  |  ph }  =  { y } }
4 euabsn 3820 . . 3  |-  ( E! y ph  <->  E. y { y  |  ph }  =  { y } )
54abbii 2500 . 2  |-  { x  |  E! y ph }  =  { x  |  E. y { y  |  ph }  =  { y } }
61, 3, 53eqtr4i 2418 1  |-  dom  F  =  { x  |  E! y ph }
Colors of variables: wff set class
Syntax hints:   E.wex 1547    = wceq 1649   E!weu 2239   {cab 2374   {csn 3758   {copab 4207   dom cdm 4819
This theorem is referenced by:  opabiota  6475
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pr 4345
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-rab 2659  df-v 2902  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-br 4155  df-opab 4209  df-dm 4829
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