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

Theorem dmopab 5929
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 5218 . . 3 𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}
2 nfopab2 5219 . . 3 𝑦{⟨𝑥, 𝑦⟩ ∣ 𝜑}
31, 2dfdmf 5910 . 2 dom {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑥 ∣ ∃𝑦 𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦}
4 df-br 5149 . . . . 5 (𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
5 opabidw 5534 . . . . 5 (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜑)
64, 5bitri 275 . . . 4 (𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦𝜑)
76exbii 1845 . . 3 (∃𝑦 𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦 ↔ ∃𝑦𝜑)
87abbii 2807 . 2 {𝑥 ∣ ∃𝑦 𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦} = {𝑥 ∣ ∃𝑦𝜑}
93, 8eqtri 2763 1 dom {⟨𝑥, 𝑦⟩ ∣ 𝜑} = {𝑥 ∣ ∃𝑦𝜑}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  wex 1776  wcel 2106  {cab 2712  cop 4637   class class class wbr 5148  {copab 5210  dom cdm 5689
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-dm 5699
This theorem is referenced by:  dmopabelb  5930  dmopabss  5932  dmopab3  5933  mptfnf  6704  opabiotadm  6990  fndmin  7065  dmoprab  7535  zfrep6  7978  hartogslem1  9580  dmttrcl  9759  rankf  9832  dfac3  10159  axdc2lem  10486  shftdm  15107  dfiso2  17820  adjeu  31918  satfdm  35354  fmla0  35367  fmlasuc0  35369
  Copyright terms: Public domain W3C validator