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Theorem args 5412
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 the alternate definition dffv4 6100 of function value, which is based on the idea in Quine's definition. (Contributed by NM, 8-May-2005.)
Assertion
Ref Expression
args {𝑥 ∣ ∃𝑦(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 3176 . . . . . 6 𝑥 ∈ V
2 imasng 5406 . . . . . 6 (𝑥 ∈ V → (𝐹 “ {𝑥}) = {𝑦𝑥𝐹𝑦})
31, 2ax-mp 5 . . . . 5 (𝐹 “ {𝑥}) = {𝑦𝑥𝐹𝑦}
43eqeq1i 2615 . . . 4 ((𝐹 “ {𝑥}) = {𝑦} ↔ {𝑦𝑥𝐹𝑦} = {𝑦})
54exbii 1764 . . 3 (∃𝑦(𝐹 “ {𝑥}) = {𝑦} ↔ ∃𝑦{𝑦𝑥𝐹𝑦} = {𝑦})
6 euabsn 4205 . . 3 (∃!𝑦 𝑥𝐹𝑦 ↔ ∃𝑦{𝑦𝑥𝐹𝑦} = {𝑦})
75, 6bitr4i 266 . 2 (∃𝑦(𝐹 “ {𝑥}) = {𝑦} ↔ ∃!𝑦 𝑥𝐹𝑦)
87abbii 2726 1 {𝑥 ∣ ∃𝑦(𝐹 “ {𝑥}) = {𝑦}} = {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1475  wex 1695  wcel 1977  ∃!weu 2458  {cab 2596  Vcvv 3173  {csn 4125   class class class wbr 4583  cima 5041
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-cnv 5046  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051
This theorem is referenced by: (None)
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