Theorem args 4816

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 2625	. . . . . 6 |- x e. _V
2		imasng 4812	. . . . . 6 |- ( x e. _V -> ( F " { x } ) = { y | x F y } )
3	1, 2	ax-mp 7	. . . . 5 |- ( F " { x } ) = { y | x F y }
4	3	eqeq1i 2096	. . . 4 |- ( ( F " { x } ) = { y } <-> { y | x F y } = { y } )
5	4	exbii 1542	. . 3 |- ( E. y ( F " { x } ) = { y } <-> E. y { y | x F y } = { y } )
6		euabsn 3518	. . 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 2204	1 |- { x | E. y ( F " { x } ) = { y } } = { x | E! y x F y }

Syntax hints:   = wceq 1290   E.wex 1427   e. wcel 1439   E!weu 1949   {cab 2075   _Vcvv 2622   {csn 3452   class class class wbr 3853   "cima 4457

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3965  ax-pow 4017  ax-pr 4047

This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2624  df-sbc 2844  df-un 3006  df-in 3008  df-ss 3015  df-pw 3437  df-sn 3458  df-pr 3459  df-op 3461  df-br 3854  df-opab 3908  df-xp 4460  df-cnv 4462  df-dm 4464  df-rn 4465  df-res 4466  df-ima 4467

This theorem is referenced by: (None)