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Theorem opabiotadm 6990
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⟩ ∣ {𝑦𝜑} = {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
StepHypRef Expression
1 dmopab 5929 . 2 dom {⟨𝑥, 𝑦⟩ ∣ {𝑦𝜑} = {𝑦}} = {𝑥 ∣ ∃𝑦{𝑦𝜑} = {𝑦}}
2 opabiota.1 . . 3 𝐹 = {⟨𝑥, 𝑦⟩ ∣ {𝑦𝜑} = {𝑦}}
32dmeqi 5918 . 2 dom 𝐹 = dom {⟨𝑥, 𝑦⟩ ∣ {𝑦𝜑} = {𝑦}}
4 euabsn 4731 . . 3 (∃!𝑦𝜑 ↔ ∃𝑦{𝑦𝜑} = {𝑦})
54abbii 2807 . 2 {𝑥 ∣ ∃!𝑦𝜑} = {𝑥 ∣ ∃𝑦{𝑦𝜑} = {𝑦}}
61, 3, 53eqtr4i 2773 1 dom 𝐹 = {𝑥 ∣ ∃!𝑦𝜑}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  wex 1776  ∃!weu 2566  {cab 2712  {csn 4631  {copab 5210  dom cdm 5689
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-dm 5699
This theorem is referenced by:  opabiota  6991
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