Theorem args 5051

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 2775	. . . . . 6 |- x e. _V
2		imasng 5047	. . . . . 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 2213	. . . 4 |- ( ( F " { x } ) = { y } <-> { y | x F y } = { y } )
5	4	exbii 1628	. . 3 |- ( E. y ( F " { x } ) = { y } <-> E. y { y | x F y } = { y } )
6		euabsn 3703	. . 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 2321	1 |- { x | E. y ( F " { x } ) = { y } } = { x | E! y x F y }

Syntax hints:   = wceq 1373   E.wex 1515   E!weu 2054   e. wcel 2176   {cab 2191   _Vcvv 2772   {csn 3633   class class class wbr 4044   "cima 4678

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-sbc 2999  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-br 4045  df-opab 4106  df-xp 4681  df-cnv 4683  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688

This theorem is referenced by: (None)