Theorem args 6078

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 6860 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 6070	. . . . . 6 ⊢ (𝑥 ∈ V → (𝐹 “ {𝑥}) = {𝑦 ∣ 𝑥𝐹𝑦})
2	1	elv 3458	. . . . 5 ⊢ (𝐹 “ {𝑥}) = {𝑦 ∣ 𝑥𝐹𝑦}
3	2	eqeq1i 2766	. . . 4 ⊢ ((𝐹 “ {𝑥}) = {𝑦} ↔ {𝑦 ∣ 𝑥𝐹𝑦} = {𝑦})
4	3	exbii 1867	. . 3 ⊢ (∃𝑦(𝐹 “ {𝑥}) = {𝑦} ↔ ∃𝑦{𝑦 ∣ 𝑥𝐹𝑦} = {𝑦})
5		euabsn 4684	. . 3 ⊢ (∃!𝑦 𝑥𝐹𝑦 ↔ ∃𝑦{𝑦 ∣ 𝑥𝐹𝑦} = {𝑦})
6	4, 5	bitr4i 280	. 2 ⊢ (∃𝑦(𝐹 “ {𝑥}) = {𝑦} ↔ ∃!𝑦 𝑥𝐹𝑦)
7	6	abbii 2828	1 ⊢ {𝑥 ∣ ∃𝑦(𝐹 “ {𝑥}) = {𝑦}} = {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦}

Syntax hints:   = wceq 1559  ∃wex 1798  ∃!weu 2594  {cab 2739  Vcvv 3453  {csn 4581   class class class wbr 5099   “ cima 5648

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-br 5100  df-opab 5162  df-xp 5651  df-cnv 5653  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658

This theorem is referenced by: (None)