Theorem opabiotadm 6943

Description: Define a function whose value is "the unique 𝑦 such that 𝜑(𝑥, 𝑦)". (Contributed by NM, 16-Nov-2013.)

Hypothesis

Ref	Expression
opabiota.1	⊢ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ {𝑦 ∣ 𝜑} = {𝑦}}

Assertion

Ref	Expression
opabiotadm	⊢ dom 𝐹 = {𝑥 ∣ ∃!𝑦𝜑}

Distinct variable group:   𝑥,𝑦,𝐹

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem opabiotadm

Step	Hyp	Ref	Expression
1		dmopab 5887	. 2 ⊢ dom {⟨𝑥, 𝑦⟩ ∣ {𝑦 ∣ 𝜑} = {𝑦}} = {𝑥 ∣ ∃𝑦{𝑦 ∣ 𝜑} = {𝑦}}
2		opabiota.1	. . 3 ⊢ 𝐹 = {⟨𝑥, 𝑦⟩ ∣ {𝑦 ∣ 𝜑} = {𝑦}}
3	2	dmeqi 5876	. 2 ⊢ dom 𝐹 = dom {⟨𝑥, 𝑦⟩ ∣ {𝑦 ∣ 𝜑} = {𝑦}}
4		euabsn 4682	. . 3 ⊢ (∃!𝑦𝜑 ↔ ∃𝑦{𝑦 ∣ 𝜑} = {𝑦})
5	4	abbii 2828	. 2 ⊢ {𝑥 ∣ ∃!𝑦𝜑} = {𝑥 ∣ ∃𝑦{𝑦 ∣ 𝜑} = {𝑦}}
6	1, 3, 5	3eqtr4i 2794	1 ⊢ dom 𝐹 = {𝑥 ∣ ∃!𝑦𝜑}

Syntax hints:   = wceq 1559  ∃wex 1798  ∃!weu 2594  {cab 2739  {csn 4579  {copab 5159  dom cdm 5643

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 5243  ax-pr 5387

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-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-br 5098  df-opab 5160  df-dm 5653

This theorem is referenced by:  opabiota  6944