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Theorem args 4621
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 {xy(𝐹 “ {x}) = {y}} = {x∃!y x𝐹y}
Distinct variable groups:   y,𝐹   x,y
Allowed substitution hint:   𝐹(x)

Proof of Theorem args
StepHypRef Expression
1 vex 2538 . . . . . 6 x V
2 imasng 4617 . . . . . 6 (x V → (𝐹 “ {x}) = {yx𝐹y})
31, 2ax-mp 7 . . . . 5 (𝐹 “ {x}) = {yx𝐹y}
43eqeq1i 2029 . . . 4 ((𝐹 “ {x}) = {y} ↔ {yx𝐹y} = {y})
54exbii 1478 . . 3 (y(𝐹 “ {x}) = {y} ↔ y{yx𝐹y} = {y})
6 euabsn 3414 . . 3 (∃!y x𝐹yy{yx𝐹y} = {y})
75, 6bitr4i 176 . 2 (y(𝐹 “ {x}) = {y} ↔ ∃!y x𝐹y)
87abbii 2135 1 {xy(𝐹 “ {x}) = {y}} = {x∃!y x𝐹y}
Colors of variables: wff set class
Syntax hints:   = wceq 1228  wex 1362   wcel 1374  ∃!weu 1882  {cab 2008  Vcvv 2535  {csn 3350   class class class wbr 3738  cima 4275
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-sbc 2742  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-br 3739  df-opab 3793  df-xp 4278  df-cnv 4280  df-dm 4282  df-rn 4283  df-res 4284  df-ima 4285
This theorem is referenced by: (None)
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