Theorem args 4714

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 2604	. . . . . 6 |- x e. _V
2		imasng 4710	. . . . . 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 2088	. . . 4 |- ( ( F " { x } ) = { y } <-> { y | x F y } = { y } )
5	4	exbii 1536	. . 3 |- ( E. y ( F " { x } ) = { y } <-> E. y { y | x F y } = { y } )
6		euabsn 3462	. . 3 |- ( E! y x F y <-> E. y { y | x F y } = { y } )
7	5, 6	bitr4i 185	. 2 |- ( E. y ( F " { x } ) = { y } <-> E! y x F y )
8	7	abbii 2194	1 |- { x | E. y ( F " { x } ) = { y } } = { x | E! y x F y }

Syntax hints:   = wceq 1284   E.wex 1421   e. wcel 1433   E!weu 1941   {cab 2067   _Vcvv 2601   {csn 3398   class class class wbr 3785   "cima 4366

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-sbc 2816  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-opab 3840  df-xp 4369  df-cnv 4371  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376

This theorem is referenced by: (None)