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Theorem opabiotadm 6422
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 5490 . 2 dom {⟨𝑥, 𝑦⟩ ∣ {𝑦𝜑} = {𝑦}} = {𝑥 ∣ ∃𝑦{𝑦𝜑} = {𝑦}}
2 opabiota.1 . . 3 𝐹 = {⟨𝑥, 𝑦⟩ ∣ {𝑦𝜑} = {𝑦}}
32dmeqi 5480 . 2 dom 𝐹 = dom {⟨𝑥, 𝑦⟩ ∣ {𝑦𝜑} = {𝑦}}
4 euabsn 4405 . . 3 (∃!𝑦𝜑 ↔ ∃𝑦{𝑦𝜑} = {𝑦})
54abbii 2877 . 2 {𝑥 ∣ ∃!𝑦𝜑} = {𝑥 ∣ ∃𝑦{𝑦𝜑} = {𝑦}}
61, 3, 53eqtr4i 2792 1 dom 𝐹 = {𝑥 ∣ ∃!𝑦𝜑}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1632  wex 1853  ∃!weu 2607  {cab 2746  {csn 4321  {copab 4864  dom cdm 5266
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-dm 5276
This theorem is referenced by:  opabiota  6423
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