Theorem args 4916

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 2692	. . . . . 6 |- x e. _V
2		imasng 4912	. . . . . 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 2148	. . . 4 |- ( ( F " { x } ) = { y } <-> { y | x F y } = { y } )
5	4	exbii 1585	. . 3 |- ( E. y ( F " { x } ) = { y } <-> E. y { y | x F y } = { y } )
6		euabsn 3601	. . 3 |- ( E! y x F y <-> E. y { y | x F y } = { y } )
7	5, 6	bitr4i 186	. 2 |- ( E. y ( F " { x } ) = { y } <-> E! y x F y )
8	7	abbii 2256	1 |- { x | E. y ( F " { x } ) = { y } } = { x | E! y x F y }

Syntax hints:   = wceq 1332   E.wex 1469   e. wcel 1481   E!weu 2000   {cab 2126   _Vcvv 2689   {csn 3532   class class class wbr 3937   "cima 4550

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-sbc 2914  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-br 3938  df-opab 3998  df-xp 4553  df-cnv 4555  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560

This theorem is referenced by: (None)