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Theorem args 5018
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 2755 . . . . . 6 𝑥 ∈ V
2 imasng 5014 . . . . . 6 (𝑥 ∈ V → (𝐹 “ {𝑥}) = {𝑦𝑥𝐹𝑦})
31, 2ax-mp 5 . . . . 5 (𝐹 “ {𝑥}) = {𝑦𝑥𝐹𝑦}
43eqeq1i 2197 . . . 4 ((𝐹 “ {𝑥}) = {𝑦} ↔ {𝑦𝑥𝐹𝑦} = {𝑦})
54exbii 1616 . . 3 (∃𝑦(𝐹 “ {𝑥}) = {𝑦} ↔ ∃𝑦{𝑦𝑥𝐹𝑦} = {𝑦})
6 euabsn 3680 . . 3 (∃!𝑦 𝑥𝐹𝑦 ↔ ∃𝑦{𝑦𝑥𝐹𝑦} = {𝑦})
75, 6bitr4i 187 . 2 (∃𝑦(𝐹 “ {𝑥}) = {𝑦} ↔ ∃!𝑦 𝑥𝐹𝑦)
87abbii 2305 1 {𝑥 ∣ ∃𝑦(𝐹 “ {𝑥}) = {𝑦}} = {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦}
Colors of variables: wff set class
Syntax hints:   = wceq 1364  wex 1503  ∃!weu 2038  wcel 2160  {cab 2175  Vcvv 2752  {csn 3610   class class class wbr 4021  cima 4650
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4139  ax-pow 4195  ax-pr 4230
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-un 3148  df-in 3150  df-ss 3157  df-pw 3595  df-sn 3616  df-pr 3617  df-op 3619  df-br 4022  df-opab 4083  df-xp 4653  df-cnv 4655  df-dm 4657  df-rn 4658  df-res 4659  df-ima 4660
This theorem is referenced by: (None)
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