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Theorem args 4994
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 4654, 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 2743 . . . . . 6  |-  x  e. 
_V
2 imasng 4988 . . . . . 6  |-  ( x  e.  _V  ->  ( F " { x }
)  =  { y  |  x F y } )
31, 2ax-mp 10 . . . . 5  |-  ( F
" { x }
)  =  { y  |  x F y }
43eqeq1i 2263 . . . 4  |-  ( ( F " { x } )  =  {
y }  <->  { y  |  x F y }  =  { y } )
54exbii 1580 . . 3  |-  ( E. y ( F " { x } )  =  { y }  <->  E. y { y  |  x F y }  =  { y } )
6 euabsn 3640 . . 3  |-  ( E! y  x F y  <->  E. y { y  |  x F y }  =  { y } )
75, 6bitr4i 245 . 2  |-  ( E. y ( F " { x } )  =  { y }  <-> 
E! y  x F y )
87abbii 2368 1  |-  { x  |  E. y ( F
" { x }
)  =  { y } }  =  {
x  |  E! y  x F y }
Colors of variables: wff set class
Syntax hints:   E.wex 1537    = wceq 1619    e. wcel 1621   E!weu 2117   {cab 2242   _Vcvv 2740   {csn 3581   class class class wbr 3963   "cima 4629
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4081  ax-nul 4089  ax-pr 4152
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2520  df-rex 2521  df-rab 2523  df-v 2742  df-sbc 2936  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-sn 3587  df-pr 3588  df-op 3590  df-br 3964  df-opab 4018  df-xp 4640  df-cnv 4642  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647
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