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Theorem args 5930
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 6640 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 imasng 5924 . . . . . 6 (𝑥 ∈ V → (𝐹 “ {𝑥}) = {𝑦𝑥𝐹𝑦})
21elv 3476 . . . . 5 (𝐹 “ {𝑥}) = {𝑦𝑥𝐹𝑦}
32eqeq1i 2826 . . . 4 ((𝐹 “ {𝑥}) = {𝑦} ↔ {𝑦𝑥𝐹𝑦} = {𝑦})
43exbii 1849 . . 3 (∃𝑦(𝐹 “ {𝑥}) = {𝑦} ↔ ∃𝑦{𝑦𝑥𝐹𝑦} = {𝑦})
5 euabsn 4635 . . 3 (∃!𝑦 𝑥𝐹𝑦 ↔ ∃𝑦{𝑦𝑥𝐹𝑦} = {𝑦})
64, 5bitr4i 281 . 2 (∃𝑦(𝐹 “ {𝑥}) = {𝑦} ↔ ∃!𝑦 𝑥𝐹𝑦)
76abbii 2886 1 {𝑥 ∣ ∃𝑦(𝐹 “ {𝑥}) = {𝑦}} = {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1538  wex 1781  ∃!weu 2653  {cab 2799  Vcvv 3471  {csn 4540   class class class wbr 5039  cima 5531
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pr 5303
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-sbc 3750  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-sn 4541  df-pr 4543  df-op 4547  df-br 5040  df-opab 5102  df-xp 5534  df-cnv 5536  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541
This theorem is referenced by: (None)
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