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Theorem opabdm 30525
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 5535 . 2 dom 𝑅 = {𝑥 ∣ ∃𝑦 𝑥𝑅𝑦}
2 nfopab1 5099 . . . 4 𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}
32nfeq2 2916 . . 3 𝑥 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}
4 nfopab2 5100 . . . . 5 𝑦{⟨𝑥, 𝑦⟩ ∣ 𝜑}
54nfeq2 2916 . . . 4 𝑦 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}
6 df-br 5031 . . . . 5 (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
7 eleq2 2821 . . . . . 6 (𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑} → (⟨𝑥, 𝑦⟩ ∈ 𝑅 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}))
8 opabidw 5380 . . . . . 6 (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜑)
97, 8bitrdi 290 . . . . 5 (𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑} → (⟨𝑥, 𝑦⟩ ∈ 𝑅𝜑))
106, 9syl5bb 286 . . . 4 (𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑} → (𝑥𝑅𝑦𝜑))
115, 10exbid 2225 . . 3 (𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑} → (∃𝑦 𝑥𝑅𝑦 ↔ ∃𝑦𝜑))
123, 11abbid 2804 . 2 (𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑} → {𝑥 ∣ ∃𝑦 𝑥𝑅𝑦} = {𝑥 ∣ ∃𝑦𝜑})
131, 12syl5eq 2785 1 (𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑} → dom 𝑅 = {𝑥 ∣ ∃𝑦𝜑})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wex 1786  wcel 2114  {cab 2716  cop 4522   class class class wbr 5030  {copab 5092  dom cdm 5525
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pr 5296
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-v 3400  df-dif 3846  df-un 3848  df-nul 4212  df-if 4415  df-sn 4517  df-pr 4519  df-op 4523  df-br 5031  df-opab 5093  df-dm 5535
This theorem is referenced by:  fpwrelmapffslem  30642
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