Theorem opabiotadm 6832

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 5813	. 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 5802	. 2 ⊢ dom 𝐹 = dom {⟨𝑥, 𝑦⟩ ∣ {𝑦 ∣ 𝜑} = {𝑦}}
4		euabsn 4659	. . 3 ⊢ (∃!𝑦𝜑 ↔ ∃𝑦{𝑦 ∣ 𝜑} = {𝑦})
5	4	abbii 2809	. 2 ⊢ {𝑥 ∣ ∃!𝑦𝜑} = {𝑥 ∣ ∃𝑦{𝑦 ∣ 𝜑} = {𝑦}}
6	1, 3, 5	3eqtr4i 2776	1 ⊢ dom 𝐹 = {𝑥 ∣ ∃!𝑦𝜑}

Syntax hints:   = wceq 1539  ∃wex 1783  ∃!weu 2568  {cab 2715  {csn 4558  {copab 5132  dom cdm 5580

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-opab 5133  df-dm 5590

This theorem is referenced by:  opabiota  6833