Users' Mathboxes Mathbox for Peter Mazsa < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  brabidga Structured version   Visualization version   GIF version

Theorem brabidga 37870
Description: The law of concretion for a binary relation. Special case of brabga 5540. Usage of this theorem is discouraged because it depends on ax-13 2366, see brabidgaw 37869 for a weaker version that does not require it. (Contributed by Peter Mazsa, 24-Nov-2018.) (New usage is discouraged.)
Hypothesis
Ref Expression
brabidga.1 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}
Assertion
Ref Expression
brabidga (𝑥𝑅𝑦𝜑)

Proof of Theorem brabidga
StepHypRef Expression
1 brabidga.1 . . 3 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}
21breqi 5158 . 2 (𝑥𝑅𝑦𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦)
3 df-br 5153 . 2 (𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
4 opabid 5531 . 2 (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜑)
52, 3, 43bitri 296 1 (𝑥𝑅𝑦𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1533  wcel 2098  cop 4638   class class class wbr 5152  {copab 5214
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2166  ax-13 2366  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-br 5153  df-opab 5215
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator