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Theorem opabiotadm 6973
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 5915 . 2 dom {⟨𝑥, 𝑦⟩ ∣ {𝑦𝜑} = {𝑦}} = {𝑥 ∣ ∃𝑦{𝑦𝜑} = {𝑦}}
2 opabiota.1 . . 3 𝐹 = {⟨𝑥, 𝑦⟩ ∣ {𝑦𝜑} = {𝑦}}
32dmeqi 5904 . 2 dom 𝐹 = dom {⟨𝑥, 𝑦⟩ ∣ {𝑦𝜑} = {𝑦}}
4 euabsn 4730 . . 3 (∃!𝑦𝜑 ↔ ∃𝑦{𝑦𝜑} = {𝑦})
54abbii 2802 . 2 {𝑥 ∣ ∃!𝑦𝜑} = {𝑥 ∣ ∃𝑦{𝑦𝜑} = {𝑦}}
61, 3, 53eqtr4i 2770 1 dom 𝐹 = {𝑥 ∣ ∃!𝑦𝜑}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1541  wex 1781  ∃!weu 2562  {cab 2709  {csn 4628  {copab 5210  dom cdm 5676
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-dm 5686
This theorem is referenced by:  opabiota  6974
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