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Theorem opabiotadm 6529
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 5072 . 2  |-  dom  { <. x ,  y >.  |  { y  |  ph }  =  { y } }  =  {
x  |  E. y { y  |  ph }  =  { y } }
2 opabiota.1 . . 3  |-  F  =  { <. x ,  y
>.  |  { y  |  ph }  =  {
y } }
32dmeqi 5063 . 2  |-  dom  F  =  dom  { <. x ,  y >.  |  {
y  |  ph }  =  { y } }
4 euabsn 3868 . . 3  |-  ( E! y ph  <->  E. y { y  |  ph }  =  { y } )
54abbii 2547 . 2  |-  { x  |  E! y ph }  =  { x  |  E. y { y  |  ph }  =  { y } }
61, 3, 53eqtr4i 2465 1  |-  dom  F  =  { x  |  E! y ph }
Colors of variables: wff set class
Syntax hints:   E.wex 1550    = wceq 1652   E!weu 2280   {cab 2421   {csn 3806   {copab 4257   dom cdm 4870
This theorem is referenced by:  opabiota  6530
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-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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