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Theorem coundi 6267
Description: Class composition distributes over union. (Contributed by NM, 21-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
coundi (𝐴 ∘ (𝐵𝐶)) = ((𝐴𝐵) ∪ (𝐴𝐶))

Proof of Theorem coundi
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 unopab 5224 . . 3 ({⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐵𝑧𝑧𝐴𝑦)} ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐶𝑧𝑧𝐴𝑦)}) = {⟨𝑥, 𝑦⟩ ∣ (∃𝑧(𝑥𝐵𝑧𝑧𝐴𝑦) ∨ ∃𝑧(𝑥𝐶𝑧𝑧𝐴𝑦))}
2 brun 5194 . . . . . . . 8 (𝑥(𝐵𝐶)𝑧 ↔ (𝑥𝐵𝑧𝑥𝐶𝑧))
32anbi1i 624 . . . . . . 7 ((𝑥(𝐵𝐶)𝑧𝑧𝐴𝑦) ↔ ((𝑥𝐵𝑧𝑥𝐶𝑧) ∧ 𝑧𝐴𝑦))
4 andir 1011 . . . . . . 7 (((𝑥𝐵𝑧𝑥𝐶𝑧) ∧ 𝑧𝐴𝑦) ↔ ((𝑥𝐵𝑧𝑧𝐴𝑦) ∨ (𝑥𝐶𝑧𝑧𝐴𝑦)))
53, 4bitri 275 . . . . . 6 ((𝑥(𝐵𝐶)𝑧𝑧𝐴𝑦) ↔ ((𝑥𝐵𝑧𝑧𝐴𝑦) ∨ (𝑥𝐶𝑧𝑧𝐴𝑦)))
65exbii 1848 . . . . 5 (∃𝑧(𝑥(𝐵𝐶)𝑧𝑧𝐴𝑦) ↔ ∃𝑧((𝑥𝐵𝑧𝑧𝐴𝑦) ∨ (𝑥𝐶𝑧𝑧𝐴𝑦)))
7 19.43 1882 . . . . 5 (∃𝑧((𝑥𝐵𝑧𝑧𝐴𝑦) ∨ (𝑥𝐶𝑧𝑧𝐴𝑦)) ↔ (∃𝑧(𝑥𝐵𝑧𝑧𝐴𝑦) ∨ ∃𝑧(𝑥𝐶𝑧𝑧𝐴𝑦)))
86, 7bitr2i 276 . . . 4 ((∃𝑧(𝑥𝐵𝑧𝑧𝐴𝑦) ∨ ∃𝑧(𝑥𝐶𝑧𝑧𝐴𝑦)) ↔ ∃𝑧(𝑥(𝐵𝐶)𝑧𝑧𝐴𝑦))
98opabbii 5210 . . 3 {⟨𝑥, 𝑦⟩ ∣ (∃𝑧(𝑥𝐵𝑧𝑧𝐴𝑦) ∨ ∃𝑧(𝑥𝐶𝑧𝑧𝐴𝑦))} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥(𝐵𝐶)𝑧𝑧𝐴𝑦)}
101, 9eqtri 2765 . 2 ({⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐵𝑧𝑧𝐴𝑦)} ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐶𝑧𝑧𝐴𝑦)}) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥(𝐵𝐶)𝑧𝑧𝐴𝑦)}
11 df-co 5694 . . 3 (𝐴𝐵) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐵𝑧𝑧𝐴𝑦)}
12 df-co 5694 . . 3 (𝐴𝐶) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐶𝑧𝑧𝐴𝑦)}
1311, 12uneq12i 4166 . 2 ((𝐴𝐵) ∪ (𝐴𝐶)) = ({⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐵𝑧𝑧𝐴𝑦)} ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐶𝑧𝑧𝐴𝑦)})
14 df-co 5694 . 2 (𝐴 ∘ (𝐵𝐶)) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥(𝐵𝐶)𝑧𝑧𝐴𝑦)}
1510, 13, 143eqtr4ri 2776 1 (𝐴 ∘ (𝐵𝐶)) = ((𝐴𝐵) ∪ (𝐴𝐶))
Colors of variables: wff setvar class
Syntax hints:  wa 395  wo 848   = wceq 1540  wex 1779  cun 3949   class class class wbr 5143  {copab 5205  ccom 5689
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
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-un 3956  df-br 5144  df-opab 5206  df-co 5694
This theorem is referenced by:  f1ofvswap  7326  mvdco  19463  ustssco  24223  coprprop  32708  cycpmconjv  33162  cvmliftlem10  35299  poimirlem9  37636  diophren  42824  rtrclex  43630  trclubgNEW  43631  trclexi  43633  rtrclexi  43634  cnvtrcl0  43639  trrelsuperrel2dg  43684  cotrclrcl  43755  frege131d  43777  dftpos6  48775
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