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Theorem opabiotadm 6974
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 5912 . 2 dom {⟨𝑥, 𝑦⟩ ∣ {𝑦𝜑} = {𝑦}} = {𝑥 ∣ ∃𝑦{𝑦𝜑} = {𝑦}}
2 opabiota.1 . . 3 𝐹 = {⟨𝑥, 𝑦⟩ ∣ {𝑦𝜑} = {𝑦}}
32dmeqi 5901 . 2 dom 𝐹 = dom {⟨𝑥, 𝑦⟩ ∣ {𝑦𝜑} = {𝑦}}
4 euabsn 4726 . . 3 (∃!𝑦𝜑 ↔ ∃𝑦{𝑦𝜑} = {𝑦})
54abbii 2795 . 2 {𝑥 ∣ ∃!𝑦𝜑} = {𝑥 ∣ ∃𝑦{𝑦𝜑} = {𝑦}}
61, 3, 53eqtr4i 2763 1 dom 𝐹 = {𝑥 ∣ ∃!𝑦𝜑}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533  wex 1773  ∃!weu 2556  {cab 2702  {csn 4624  {copab 5205  dom cdm 5672
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-rab 3420  df-v 3465  df-dif 3943  df-un 3945  df-ss 3957  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5144  df-opab 5206  df-dm 5682
This theorem is referenced by:  opabiota  6975
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