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Theorem opabdm 29722
Description: Domain of an ordered-pair class abstraction. (Contributed by Thierry Arnoux, 31-Aug-2017.)
Assertion
Ref Expression
opabdm (𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑} → dom 𝑅 = {𝑥 ∣ ∃𝑦𝜑})
Distinct variable group:   𝑥,𝑦,𝑅
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem opabdm
StepHypRef Expression
1 df-dm 5268 . 2 dom 𝑅 = {𝑥 ∣ ∃𝑦 𝑥𝑅𝑦}
2 nfopab1 4863 . . . 4 𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}
32nfeq2 2910 . . 3 𝑥 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}
4 nfopab2 4864 . . . . 5 𝑦{⟨𝑥, 𝑦⟩ ∣ 𝜑}
54nfeq2 2910 . . . 4 𝑦 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}
6 df-br 4797 . . . . 5 (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
7 eleq2 2820 . . . . . 6 (𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑} → (⟨𝑥, 𝑦⟩ ∈ 𝑅 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}))
8 opabid 5124 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜑)
97, 8syl6bb 276 . . . . 5 (𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑} → (⟨𝑥, 𝑦⟩ ∈ 𝑅𝜑))
106, 9syl5bb 272 . . . 4 (𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑} → (𝑥𝑅𝑦𝜑))
115, 10exbid 2230 . . 3 (𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑} → (∃𝑦 𝑥𝑅𝑦 ↔ ∃𝑦𝜑))
123, 11abbid 2870 . 2 (𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑} → {𝑥 ∣ ∃𝑦 𝑥𝑅𝑦} = {𝑥 ∣ ∃𝑦𝜑})
131, 12syl5eq 2798 1 (𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑} → dom 𝑅 = {𝑥 ∣ ∃𝑦𝜑})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1624  wex 1845  wcel 2131  {cab 2738  cop 4319   class class class wbr 4796  {copab 4856  dom cdm 5258
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pr 5047
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-rab 3051  df-v 3334  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-sn 4314  df-pr 4316  df-op 4320  df-br 4797  df-opab 4857  df-dm 5268
This theorem is referenced by:  fpwrelmapffslem  29808
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