Theorem args 5056

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 2776	. . . . . 6 ⊢ 𝑥 ∈ V
2		imasng 5052	. . . . . 6 ⊢ (𝑥 ∈ V → (𝐹 “ {𝑥}) = {𝑦 ∣ 𝑥𝐹𝑦})
3	1, 2	ax-mp 5	. . . . 5 ⊢ (𝐹 “ {𝑥}) = {𝑦 ∣ 𝑥𝐹𝑦}
4	3	eqeq1i 2214	. . . 4 ⊢ ((𝐹 “ {𝑥}) = {𝑦} ↔ {𝑦 ∣ 𝑥𝐹𝑦} = {𝑦})
5	4	exbii 1629	. . 3 ⊢ (∃𝑦(𝐹 “ {𝑥}) = {𝑦} ↔ ∃𝑦{𝑦 ∣ 𝑥𝐹𝑦} = {𝑦})
6		euabsn 3704	. . 3 ⊢ (∃!𝑦 𝑥𝐹𝑦 ↔ ∃𝑦{𝑦 ∣ 𝑥𝐹𝑦} = {𝑦})
7	5, 6	bitr4i 187	. 2 ⊢ (∃𝑦(𝐹 “ {𝑥}) = {𝑦} ↔ ∃!𝑦 𝑥𝐹𝑦)
8	7	abbii 2322	1 ⊢ {𝑥 ∣ ∃𝑦(𝐹 “ {𝑥}) = {𝑦}} = {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦}

Syntax hints:   = wceq 1373  ∃wex 1516  ∃!weu 2055   ∈ wcel 2177  {cab 2192  Vcvv 2773  {csn 3634   class class class wbr 4047   “ cima 4682

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2180  ax-ext 2188  ax-sep 4166  ax-pow 4222  ax-pr 4257

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-v 2775  df-sbc 3000  df-un 3171  df-in 3173  df-ss 3180  df-pw 3619  df-sn 3640  df-pr 3641  df-op 3643  df-br 4048  df-opab 4110  df-xp 4685  df-cnv 4687  df-dm 4689  df-rn 4690  df-res 4691  df-ima 4692

This theorem is referenced by: (None)