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Theorem args 5041
Description: Two ways to express the class of unique-valued arguments of 
F, which is the same as the domain of  F whenever  F is a function. The left-hand side of the equality is from Definition 10.2 of [Quine] p. 65. Quine uses the notation "arg  F " for this class (for which we have no separate notation). Observe the resemblance to our df-fv 5263, which was based on the idea in Quine's definition. (Contributed by NM, 8-May-2005.)
Assertion
Ref Expression
args  |-  { x  |  E. y ( F
" { x }
)  =  { y } }  =  {
x  |  E! y  x F y }
Distinct variable groups:    y, F    x, y
Allowed substitution hint:    F( x)

Proof of Theorem args
StepHypRef Expression
1 vex 2791 . . . . . 6  |-  x  e. 
_V
2 imasng 5035 . . . . . 6  |-  ( x  e.  _V  ->  ( F " { x }
)  =  { y  |  x F y } )
31, 2ax-mp 8 . . . . 5  |-  ( F
" { x }
)  =  { y  |  x F y }
43eqeq1i 2290 . . . 4  |-  ( ( F " { x } )  =  {
y }  <->  { y  |  x F y }  =  { y } )
54exbii 1569 . . 3  |-  ( E. y ( F " { x } )  =  { y }  <->  E. y { y  |  x F y }  =  { y } )
6 euabsn 3699 . . 3  |-  ( E! y  x F y  <->  E. y { y  |  x F y }  =  { y } )
75, 6bitr4i 243 . 2  |-  ( E. y ( F " { x } )  =  { y }  <-> 
E! y  x F y )
87abbii 2395 1  |-  { x  |  E. y ( F
" { x }
)  =  { y } }  =  {
x  |  E! y  x F y }
Colors of variables: wff set class
Syntax hints:   E.wex 1528    = wceq 1623    e. wcel 1684   E!weu 2143   {cab 2269   _Vcvv 2788   {csn 3640   class class class wbr 4023   "cima 4692
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-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  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-xp 4695  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702
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