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Theorem args 4844
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 {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 2644 . . . . . 6 𝑥 ∈ V
2 imasng 4840 . . . . . 6 (𝑥 ∈ V → (𝐹 “ {𝑥}) = {𝑦𝑥𝐹𝑦})
31, 2ax-mp 7 . . . . 5 (𝐹 “ {𝑥}) = {𝑦𝑥𝐹𝑦}
43eqeq1i 2107 . . . 4 ((𝐹 “ {𝑥}) = {𝑦} ↔ {𝑦𝑥𝐹𝑦} = {𝑦})
54exbii 1552 . . 3 (∃𝑦(𝐹 “ {𝑥}) = {𝑦} ↔ ∃𝑦{𝑦𝑥𝐹𝑦} = {𝑦})
6 euabsn 3540 . . 3 (∃!𝑦 𝑥𝐹𝑦 ↔ ∃𝑦{𝑦𝑥𝐹𝑦} = {𝑦})
75, 6bitr4i 186 . 2 (∃𝑦(𝐹 “ {𝑥}) = {𝑦} ↔ ∃!𝑦 𝑥𝐹𝑦)
87abbii 2215 1 {𝑥 ∣ ∃𝑦(𝐹 “ {𝑥}) = {𝑦}} = {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦}
Colors of variables: wff set class
Syntax hints:   = wceq 1299  wex 1436  wcel 1448  ∃!weu 1960  {cab 2086  Vcvv 2641  {csn 3474   class class class wbr 3875  cima 4480
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069
This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-v 2643  df-sbc 2863  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-br 3876  df-opab 3930  df-xp 4483  df-cnv 4485  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490
This theorem is referenced by: (None)
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