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Theorem args 4694
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). (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 2560 . . . . . 6  |-  x  e. 
_V
2 imasng 4690 . . . . . 6  |-  ( x  e.  _V  ->  ( F " { x }
)  =  { y  |  x F y } )
31, 2ax-mp 7 . . . . 5  |-  ( F
" { x }
)  =  { y  |  x F y }
43eqeq1i 2047 . . . 4  |-  ( ( F " { x } )  =  {
y }  <->  { y  |  x F y }  =  { y } )
54exbii 1496 . . 3  |-  ( E. y ( F " { x } )  =  { y }  <->  E. y { y  |  x F y }  =  { y } )
6 euabsn 3440 . . 3  |-  ( E! y  x F y  <->  E. y { y  |  x F y }  =  { y } )
75, 6bitr4i 176 . 2  |-  ( E. y ( F " { x } )  =  { y }  <-> 
E! y  x F y )
87abbii 2153 1  |-  { x  |  E. y ( F
" { x }
)  =  { y } }  =  {
x  |  E! y  x F y }
Colors of variables: wff set class
Syntax hints:    = wceq 1243   E.wex 1381    e. wcel 1393   E!weu 1900   {cab 2026   _Vcvv 2557   {csn 3375   class class class wbr 3764   "cima 4348
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-cnv 4353  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358
This theorem is referenced by: (None)
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