Theorem args 5370

Description: Two ways to express the class of unique-valued arguments of , which is the same as the domain of whenever is a function. The left-hand side of the equality is from Definition 10.2 of [Quine] p. 65. Quine uses the notation "arg " for this class (for which we have no separate notation). Observe the resemblance to the alternative definition dffv4 5868 of function value, which is based on the idea in Quine's definition. (Contributed by NM, 8-May-2005.)

Assertion

Ref	Expression
args

Distinct variable groups:   ,   ,

Proof of Theorem args

Step	Hyp	Ref	Expression
1		vex 3112	. . . . . 6
2		imasng 5364	. . . . . 6
3	1, 2	ax-mp 5	. . . . 5
4	3	eqeq1i 2464	. . . 4
5	4	exbii 1667	. . 3
6		euabsn 4102	. . 3
7	5, 6	bitr4i 252	. 2
8	7	abbii 2591	1

Syntax hints:  =wceq 1395  E.wex 1612  e.wcel 1818  E!weu 2282  {cab 2442  cvv 3109  {csn 4029   class class class wbr 4452  "cima 5007

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-br 4453  df-opab 4511  df-xp 5010  df-cnv 5012  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017