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Theorem opabiotadm 6227
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 5305 . 2 dom {⟨𝑥, 𝑦⟩ ∣ {𝑦𝜑} = {𝑦}} = {𝑥 ∣ ∃𝑦{𝑦𝜑} = {𝑦}}
2 opabiota.1 . . 3 𝐹 = {⟨𝑥, 𝑦⟩ ∣ {𝑦𝜑} = {𝑦}}
32dmeqi 5295 . 2 dom 𝐹 = dom {⟨𝑥, 𝑦⟩ ∣ {𝑦𝜑} = {𝑦}}
4 euabsn 4238 . . 3 (∃!𝑦𝜑 ↔ ∃𝑦{𝑦𝜑} = {𝑦})
54abbii 2736 . 2 {𝑥 ∣ ∃!𝑦𝜑} = {𝑥 ∣ ∃𝑦{𝑦𝜑} = {𝑦}}
61, 3, 53eqtr4i 2653 1 dom 𝐹 = {𝑥 ∣ ∃!𝑦𝜑}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1480  wex 1701  ∃!weu 2469  {cab 2607  {csn 4155  {copab 4682  dom cdm 5084
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-br 4624  df-opab 4684  df-dm 5094
This theorem is referenced by:  opabiota  6228
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