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 36225
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}
21relopabiv 5780 . . . 4 Rel 𝑅
32brrelex1i 5692 . . 3 (𝐴𝑅𝐵𝐴 ∈ V)
4 brabg2.1 . . . . . . 7 (𝑥 = 𝐴 → (𝜑𝜓))
5 brabg2.2 . . . . . . 7 (𝑦 = 𝐵 → (𝜓𝜒))
64, 5, 1brabg 5500 . . . . . 6 ((𝐴 ∈ V ∧ 𝐵𝐷) → (𝐴𝑅𝐵𝜒))
76biimpd 228 . . . . 5 ((𝐴 ∈ V ∧ 𝐵𝐷) → (𝐴𝑅𝐵𝜒))
87ex 414 . . . 4 (𝐴 ∈ V → (𝐵𝐷 → (𝐴𝑅𝐵𝜒)))
98com3l 89 . . 3 (𝐵𝐷 → (𝐴𝑅𝐵 → (𝐴 ∈ V → 𝜒)))
103, 9mpdi 45 . 2 (𝐵𝐷 → (𝐴𝑅𝐵𝜒))
11 brabg2.4 . . 3 (𝜒𝐴𝐶)
124, 5, 1brabg 5500 . . . . 5 ((𝐴𝐶𝐵𝐷) → (𝐴𝑅𝐵𝜒))
1312exbiri 810 . . . 4 (𝐴𝐶 → (𝐵𝐷 → (𝜒𝐴𝑅𝐵)))
1413com3l 89 . . 3 (𝐵𝐷 → (𝜒 → (𝐴𝐶𝐴𝑅𝐵)))
1511, 14mpdi 45 . 2 (𝐵𝐷 → (𝜒𝐴𝑅𝐵))
1610, 15impbid 211 1 (𝐵𝐷 → (𝐴𝑅𝐵𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  Vcvv 3447   class class class wbr 5109  {copab 5171
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-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
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-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-br 5110  df-opab 5172  df-xp 5643  df-rel 5644
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator