Theorem args 4293

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). Observe the resemblance to our df-fv 4014, which was based on the idea in Quine's definition.

Assertion

Ref	Expression
args	|- {x | E.y(F"{x}) = {y}} = {x | E!y xFy}

Distinct variable groups:   y,F   x,y

Proof of Theorem args

Step	Hyp	Ref	Expression
1		visset 2295	. . . . . 6 |- x e. _V
2		imasng 4287	. . . . . 6 |- (x e. _V -> (F"{x}) = {y | xFy})
3	1, 2	ax-mp 7	. . . . 5 |- (F"{x}) = {y | xFy}
4	3	eqeq1i 1891	. . . 4 |- ((F"{x}) = {y} <-> {y | xFy} = {y})
5	4	exbii 1398	. . 3 |- (E.y(F"{x}) = {y} <-> E.y{y | xFy} = {y})
6		euabsn 3095	. . 3 |- (E!y xFy <-> E.y{y | xFy} = {y})
7	5, 6	bitr4i 193	. 2 |- (E.y(F"{x}) = {y} <-> E!y xFy)
8	7	abbii 2006	1 |- {x | E.y(F"{x}) = {y}} = {x | E!y xFy}

Syntax hints:   = wceq 1298   e. wcel 1300  E.wex 1326  E!weu 1771  {cab 1871  _Vcvv 2292  {csn 3044   class class class wbr 3338  "cima 3989

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-br 3339  df-opab 3396  df-xp 4000  df-cnv 4002  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007

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