Theorem args 5050

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

Step	Hyp	Ref	Expression
1		vex 2774	. . . . . 6 |- x e. _V
2		imasng 5046	. . . . . 6 |- ( x e. _V -> ( F " { x } ) = { y | x F y } )
3	1, 2	ax-mp 5	. . . . 5 |- ( F " { x } ) = { y | x F y }
4	3	eqeq1i 2212	. . . 4 |- ( ( F " { x } ) = { y } <-> { y | x F y } = { y } )
5	4	exbii 1627	. . 3 |- ( E. y ( F " { x } ) = { y } <-> E. y { y | x F y } = { y } )
6		euabsn 3702	. . 3 |- ( E! y x F y <-> E. y { y | x F y } = { y } )
7	5, 6	bitr4i 187	. 2 |- ( E. y ( F " { x } ) = { y } <-> E! y x F y )
8	7	abbii 2320	1 |- { x | E. y ( F " { x } ) = { y } } = { x | E! y x F y }

Syntax hints:   = wceq 1372   E.wex 1514   E!weu 2053   e. wcel 2175   {cab 2190   _Vcvv 2771   {csn 3632   class class class wbr 4043   "cima 4677

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-v 2773  df-sbc 2998  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-br 4044  df-opab 4105  df-xp 4680  df-cnv 4682  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687

This theorem is referenced by: (None)