Theorem args 5105

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

Step	Hyp	Ref	Expression
1		vex 2805	. . . . . 6 ⊢ 𝑥 ∈ V
2		imasng 5101	. . . . . 6 ⊢ (𝑥 ∈ V → (𝐹 “ {𝑥}) = {𝑦 ∣ 𝑥𝐹𝑦})
3	1, 2	ax-mp 5	. . . . 5 ⊢ (𝐹 “ {𝑥}) = {𝑦 ∣ 𝑥𝐹𝑦}
4	3	eqeq1i 2239	. . . 4 ⊢ ((𝐹 “ {𝑥}) = {𝑦} ↔ {𝑦 ∣ 𝑥𝐹𝑦} = {𝑦})
5	4	exbii 1653	. . 3 ⊢ (∃𝑦(𝐹 “ {𝑥}) = {𝑦} ↔ ∃𝑦{𝑦 ∣ 𝑥𝐹𝑦} = {𝑦})
6		euabsn 3741	. . 3 ⊢ (∃!𝑦 𝑥𝐹𝑦 ↔ ∃𝑦{𝑦 ∣ 𝑥𝐹𝑦} = {𝑦})
7	5, 6	bitr4i 187	. 2 ⊢ (∃𝑦(𝐹 “ {𝑥}) = {𝑦} ↔ ∃!𝑦 𝑥𝐹𝑦)
8	7	abbii 2347	1 ⊢ {𝑥 ∣ ∃𝑦(𝐹 “ {𝑥}) = {𝑦}} = {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦}

Syntax hints:   = wceq 1397  ∃wex 1540  ∃!weu 2079   ∈ wcel 2202  {cab 2217  Vcvv 2802  {csn 3669   class class class wbr 4088   “ cima 4728

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151  df-xp 4731  df-cnv 4733  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738

This theorem is referenced by: (None)