ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  args Unicode version

Theorem args 4722
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 2577 . . . . . 6  |-  x  e. 
_V
2 imasng 4718 . . . . . 6  |-  ( x  e.  _V  ->  ( F " { x }
)  =  { y  |  x F y } )
31, 2ax-mp 7 . . . . 5  |-  ( F
" { x }
)  =  { y  |  x F y }
43eqeq1i 2063 . . . 4  |-  ( ( F " { x } )  =  {
y }  <->  { y  |  x F y }  =  { y } )
54exbii 1512 . . 3  |-  ( E. y ( F " { x } )  =  { y }  <->  E. y { y  |  x F y }  =  { y } )
6 euabsn 3468 . . 3  |-  ( E! y  x F y  <->  E. y { y  |  x F y }  =  { y } )
75, 6bitr4i 180 . 2  |-  ( E. y ( F " { x } )  =  { y }  <-> 
E! y  x F y )
87abbii 2169 1  |-  { x  |  E. y ( F
" { x }
)  =  { y } }  =  {
x  |  E! y  x F y }
Colors of variables: wff set class
Syntax hints:    = wceq 1259   E.wex 1397    e. wcel 1409   E!weu 1916   {cab 2042   _Vcvv 2574   {csn 3403   class class class wbr 3792   "cima 4376
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-sbc 2788  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-opab 3847  df-xp 4379  df-cnv 4381  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator