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Theorem opabiotadm 6292
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 4889 . 2  |-  dom  { <. x ,  y >.  |  { y  |  ph }  =  { y } }  =  {
x  |  E. y { y  |  ph }  =  { y } }
2 opabiota.1 . . 3  |-  F  =  { <. x ,  y
>.  |  { y  |  ph }  =  {
y } }
32dmeqi 4880 . 2  |-  dom  F  =  dom  { <. x ,  y >.  |  {
y  |  ph }  =  { y } }
4 euabsn 3699 . . 3  |-  ( E! y ph  <->  E. y { y  |  ph }  =  { y } )
54abbii 2395 . 2  |-  { x  |  E! y ph }  =  { x  |  E. y { y  |  ph }  =  { y } }
61, 3, 53eqtr4i 2313 1  |-  dom  F  =  { x  |  E! y ph }
Colors of variables: wff set class
Syntax hints:   E.wex 1528    = wceq 1623   E!weu 2143   {cab 2269   {csn 3640   {copab 4076   dom cdm 4689
This theorem is referenced by:  opabiota  6293
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-dm 4699
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