Theorem args 3428

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 3198, 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 1813	. . . . . 6 |- x e. V
2		imasng 3424	. . . . . 6 |- (x e. V -> (F"{x}) = {y | xFy})
3	1, 2	ax-mp 7	. . . . 5 |- (F"{x}) = {y | xFy}
4	3	eqeq1i 1482	. . . 4 |- ((F"{x}) = {y} <-> {y | xFy} = {y})
5	4	exbii 1051	. . 3 |- (E.y(F"{x}) = {y} <-> E.y{y | xFy} = {y})
6		eusn 2446	. . 3 |- (E!y xFy <-> E.y{y | xFy} = {y})
7	5, 6	bitr4 176	. 2 |- (E.y(F"{x}) = {y} <-> E!y xFy)
8	7	abbii 1575	1 |- {x | E.y(F"{x}) = {y}} = {x | E!y xFy}

Syntax hints:   = wceq 956   e. wcel 958  E.wex 980  E!weu 1380  {cab 1463  Vcvv 1811  {csn 2409   class class class wbr 2619  "cima 3173

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620  df-opab 2667  df-xp 3184  df-cnv 3186  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191

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