MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dmopab Structured version   Visualization version   GIF version

Theorem dmopab 5872
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 5176 . . 3 𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}
2 nfopab2 5177 . . 3 𝑦{⟨𝑥, 𝑦⟩ ∣ 𝜑}
31, 2dfdmf 5853 . 2 dom {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑥 ∣ ∃𝑦 𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦}
4 df-br 5107 . . . . 5 (𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
5 opabidw 5482 . . . . 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 4593   class class class wbr 5106  {copab 5168  dom cdm 5634
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 5257  ax-nul 5264  ax-pr 5385
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 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-dm 5644
This theorem is referenced by:  dmopabelb  5873  dmopabss  5875  dmopab3  5876  mptfnf  6637  opabiotadm  6924  fndmin  6996  dmoprab  7459  zfrep6  7888  hartogslem1  9483  dmttrcl  9662  rankf  9735  dfac3  10062  axdc2lem  10389  shftdm  14962  dfiso2  17660  adjeu  30873  satfdm  34020  fmla0  34033  fmlasuc0  34035
  Copyright terms: Public domain W3C validator