Users' Mathboxes Mathbox for Jeff Madsen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  brabg2 Structured version   Visualization version   GIF version

Theorem brabg2 32680
Description: Relation by a binary relation abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.)
Hypotheses
Ref Expression
brabg2.1 (𝑥 = 𝐴 → (𝜑𝜓))
brabg2.2 (𝑦 = 𝐵 → (𝜓𝜒))
brabg2.3 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}
brabg2.4 (𝜒𝐴𝐶)
Assertion
Ref Expression
brabg2 (𝐵𝐷 → (𝐴𝑅𝐵𝜒))
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝜒,𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝐶(𝑥,𝑦)   𝐷(𝑥,𝑦)   𝑅(𝑥,𝑦)

Proof of Theorem brabg2
StepHypRef Expression
1 brabg2.3 . . . . 5 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}
21relopabi 5167 . . . 4 Rel 𝑅
32brrelexi 5082 . . 3 (𝐴𝑅𝐵𝐴 ∈ V)
4 brabg2.1 . . . . . . 7 (𝑥 = 𝐴 → (𝜑𝜓))
5 brabg2.2 . . . . . . 7 (𝑦 = 𝐵 → (𝜓𝜒))
64, 5, 1brabg 4919 . . . . . 6 ((𝐴 ∈ V ∧ 𝐵𝐷) → (𝐴𝑅𝐵𝜒))
76biimpd 218 . . . . 5 ((𝐴 ∈ V ∧ 𝐵𝐷) → (𝐴𝑅𝐵𝜒))
87ex 449 . . . 4 (𝐴 ∈ V → (𝐵𝐷 → (𝐴𝑅𝐵𝜒)))
98com3l 87 . . 3 (𝐵𝐷 → (𝐴𝑅𝐵 → (𝐴 ∈ V → 𝜒)))
103, 9mpdi 44 . 2 (𝐵𝐷 → (𝐴𝑅𝐵𝜒))
11 brabg2.4 . . 3 (𝜒𝐴𝐶)
124, 5, 1brabg 4919 . . . . 5 ((𝐴𝐶𝐵𝐷) → (𝐴𝑅𝐵𝜒))
1312exbiri 650 . . . 4 (𝐴𝐶 → (𝐵𝐷 → (𝜒𝐴𝑅𝐵)))
1413com3l 87 . . 3 (𝐵𝐷 → (𝜒 → (𝐴𝐶𝐴𝑅𝐵)))
1511, 14mpdi 44 . 2 (𝐵𝐷 → (𝜒𝐴𝑅𝐵))
1610, 15impbid 201 1 (𝐵𝐷 → (𝐴𝑅𝐵𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383   = wceq 1475  wcel 1977  Vcvv 3173   class class class wbr 4583  {copab 4642
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator