Theorem args 6041

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 6819 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

Step	Hyp	Ref	Expression
1		imasng 6033	. . . . . 6 ⊢ (𝑥 ∈ V → (𝐹 “ {𝑥}) = {𝑦 ∣ 𝑥𝐹𝑦})
2	1	elv 3441	. . . . 5 ⊢ (𝐹 “ {𝑥}) = {𝑦 ∣ 𝑥𝐹𝑦}
3	2	eqeq1i 2736	. . . 4 ⊢ ((𝐹 “ {𝑥}) = {𝑦} ↔ {𝑦 ∣ 𝑥𝐹𝑦} = {𝑦})
4	3	exbii 1849	. . 3 ⊢ (∃𝑦(𝐹 “ {𝑥}) = {𝑦} ↔ ∃𝑦{𝑦 ∣ 𝑥𝐹𝑦} = {𝑦})
5		euabsn 4679	. . 3 ⊢ (∃!𝑦 𝑥𝐹𝑦 ↔ ∃𝑦{𝑦 ∣ 𝑥𝐹𝑦} = {𝑦})
6	4, 5	bitr4i 278	. 2 ⊢ (∃𝑦(𝐹 “ {𝑥}) = {𝑦} ↔ ∃!𝑦 𝑥𝐹𝑦)
7	6	abbii 2798	1 ⊢ {𝑥 ∣ ∃𝑦(𝐹 “ {𝑥}) = {𝑦}} = {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦}

Syntax hints:   = wceq 1541  ∃wex 1780  ∃!weu 2563  {cab 2709  Vcvv 3436  {csn 4576   class class class wbr 5091   “ cima 5619

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-br 5092  df-opab 5154  df-xp 5622  df-cnv 5624  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629

This theorem is referenced by: (None)