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Theorem dmopab 5916
Description: The domain of a class of ordered pairs. (Contributed by NM, 16-May-1995.) (Revised by Mario Carneiro, 4-Dec-2016.)
Assertion
Ref Expression
dmopab dom {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑥 ∣ ∃𝑦𝜑}
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem dmopab
StepHypRef Expression
1 nfopab1 5219 . . 3 𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}
2 nfopab2 5220 . . 3 𝑦{⟨𝑥, 𝑦⟩ ∣ 𝜑}
31, 2dfdmf 5897 . 2 dom {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑥 ∣ ∃𝑦 𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦}
4 df-br 5150 . . . . 5 (𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
5 opabidw 5525 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜑)
64, 5bitri 275 . . . 4 (𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦𝜑)
76exbii 1851 . . 3 (∃𝑦 𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦 ↔ ∃𝑦𝜑)
87abbii 2803 . 2 {𝑥 ∣ ∃𝑦 𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦} = {𝑥 ∣ ∃𝑦𝜑}
93, 8eqtri 2761 1 dom {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑥 ∣ ∃𝑦𝜑}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  wex 1782  wcel 2107  {cab 2710  cop 4635   class class class wbr 5149  {copab 5211  dom cdm 5677
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-dm 5687
This theorem is referenced by:  dmopabelb  5917  dmopabss  5919  dmopab3  5920  mptfnf  6686  opabiotadm  6974  fndmin  7047  dmoprab  7510  zfrep6  7941  hartogslem1  9537  dmttrcl  9716  rankf  9789  dfac3  10116  axdc2lem  10443  shftdm  15018  dfiso2  17719  adjeu  31142  satfdm  34360  fmla0  34373  fmlasuc0  34375
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