 Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  args Structured version   Visualization version   GIF version

Theorem args 5528
 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 6226 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 3234 . . . . . 6 𝑥 ∈ V
2 imasng 5522 . . . . . 6 (𝑥 ∈ V → (𝐹 “ {𝑥}) = {𝑦𝑥𝐹𝑦})
31, 2ax-mp 5 . . . . 5 (𝐹 “ {𝑥}) = {𝑦𝑥𝐹𝑦}
43eqeq1i 2656 . . . 4 ((𝐹 “ {𝑥}) = {𝑦} ↔ {𝑦𝑥𝐹𝑦} = {𝑦})
54exbii 1814 . . 3 (∃𝑦(𝐹 “ {𝑥}) = {𝑦} ↔ ∃𝑦{𝑦𝑥𝐹𝑦} = {𝑦})
6 euabsn 4293 . . 3 (∃!𝑦 𝑥𝐹𝑦 ↔ ∃𝑦{𝑦𝑥𝐹𝑦} = {𝑦})
75, 6bitr4i 267 . 2 (∃𝑦(𝐹 “ {𝑥}) = {𝑦} ↔ ∃!𝑦 𝑥𝐹𝑦)
87abbii 2768 1 {𝑥 ∣ ∃𝑦(𝐹 “ {𝑥}) = {𝑦}} = {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦}
 Colors of variables: wff setvar class Syntax hints:   = wceq 1523  ∃wex 1744   ∈ wcel 2030  ∃!weu 2498  {cab 2637  Vcvv 3231  {csn 4210   class class class wbr 4685   “ cima 5146 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-xp 5149  df-cnv 5151  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator