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Theorem args 4637
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 2554 . . . . . 6 x V
2 imasng 4633 . . . . . 6 (x V → (𝐹 “ {x}) = {yx𝐹y})
31, 2ax-mp 7 . . . . 5 (𝐹 “ {x}) = {yx𝐹y}
43eqeq1i 2044 . . . 4 ((𝐹 “ {x}) = {y} ↔ {yx𝐹y} = {y})
54exbii 1493 . . 3 (y(𝐹 “ {x}) = {y} ↔ y{yx𝐹y} = {y})
6 euabsn 3431 . . 3 (∃!y x𝐹yy{yx𝐹y} = {y})
75, 6bitr4i 176 . 2 (y(𝐹 “ {x}) = {y} ↔ ∃!y x𝐹y)
87abbii 2150 1 {xy(𝐹 “ {x}) = {y}} = {x∃!y x𝐹y}
Colors of variables: wff set class
Syntax hints:   = wceq 1242  wex 1378   wcel 1390  ∃!weu 1897  {cab 2023  Vcvv 2551  {csn 3367   class class class wbr 3755  cima 4291
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-cnv 4296  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301
This theorem is referenced by: (None)
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