MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  brab2a Structured version   Visualization version   GIF version

Theorem brab2a 5779
Description: The law of concretion for a binary relation. Ordered pair membership in an ordered pair class abstraction. (Contributed by Mario Carneiro, 28-Apr-2015.)
Hypotheses
Ref Expression
brab2a.1 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))
brab2a.2 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐶𝑦𝐷) ∧ 𝜑)}
Assertion
Ref Expression
brab2a (𝐴𝑅𝐵 ↔ ((𝐴𝐶𝐵𝐷) ∧ 𝜓))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦   𝑥,𝐷,𝑦   𝜓,𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝑅(𝑥,𝑦)

Proof of Theorem brab2a
StepHypRef Expression
1 brab2a.2 . . . 4 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐶𝑦𝐷) ∧ 𝜑)}
2 opabssxp 5778 . . . 4 {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐶𝑦𝐷) ∧ 𝜑)} ⊆ (𝐶 × 𝐷)
31, 2eqsstri 4030 . . 3 𝑅 ⊆ (𝐶 × 𝐷)
43brel 5750 . 2 (𝐴𝑅𝐵 → (𝐴𝐶𝐵𝐷))
5 df-br 5144 . . . 4 (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
61eleq2i 2833 . . . 4 (⟨𝐴, 𝐵⟩ ∈ 𝑅 ↔ ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐶𝑦𝐷) ∧ 𝜑)})
75, 6bitri 275 . . 3 (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐶𝑦𝐷) ∧ 𝜑)})
8 brab2a.1 . . . 4 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))
98opelopab2a 5540 . . 3 ((𝐴𝐶𝐵𝐷) → (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐶𝑦𝐷) ∧ 𝜑)} ↔ 𝜓))
107, 9bitrid 283 . 2 ((𝐴𝐶𝐵𝐷) → (𝐴𝑅𝐵𝜓))
114, 10biadanii 822 1 (𝐴𝑅𝐵 ↔ ((𝐴𝐶𝐵𝐷) ∧ 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  cop 4632   class class class wbr 5143  {copab 5205   × cxp 5683
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691
This theorem is referenced by:  fnse  8158  ltxrlt  11331  ltxr  13157  issect  17797  gaorb  19325  ispgp  19610  efgcpbllema  19772  lmbr  23266  isphtpc  25026  vitalilem1  25643  vitalilem2  25644  vitalilem3  25645  tgjustf  28481  iscgrg  28520  ishlg  28610  iscgra  28817  isinag  28846  isleag  28855  mgcval  32977  filnetlem1  36379  weiunlem1  36463  bj-brab2a1  37150  prjsprel  42614
  Copyright terms: Public domain W3C validator