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Theorem coundi 5982
 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 5046 . . 3 ({⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐵𝑧𝑧𝐴𝑦)} ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐶𝑧𝑧𝐴𝑦)}) = {⟨𝑥, 𝑦⟩ ∣ (∃𝑧(𝑥𝐵𝑧𝑧𝐴𝑦) ∨ ∃𝑧(𝑥𝐶𝑧𝑧𝐴𝑦))}
2 brun 5019 . . . . . . . 8 (𝑥(𝐵𝐶)𝑧 ↔ (𝑥𝐵𝑧𝑥𝐶𝑧))
32anbi1i 623 . . . . . . 7 ((𝑥(𝐵𝐶)𝑧𝑧𝐴𝑦) ↔ ((𝑥𝐵𝑧𝑥𝐶𝑧) ∧ 𝑧𝐴𝑦))
4 andir 1003 . . . . . . 7 (((𝑥𝐵𝑧𝑥𝐶𝑧) ∧ 𝑧𝐴𝑦) ↔ ((𝑥𝐵𝑧𝑧𝐴𝑦) ∨ (𝑥𝐶𝑧𝑧𝐴𝑦)))
53, 4bitri 276 . . . . . 6 ((𝑥(𝐵𝐶)𝑧𝑧𝐴𝑦) ↔ ((𝑥𝐵𝑧𝑧𝐴𝑦) ∨ (𝑥𝐶𝑧𝑧𝐴𝑦)))
65exbii 1833 . . . . 5 (∃𝑧(𝑥(𝐵𝐶)𝑧𝑧𝐴𝑦) ↔ ∃𝑧((𝑥𝐵𝑧𝑧𝐴𝑦) ∨ (𝑥𝐶𝑧𝑧𝐴𝑦)))
7 19.43 1868 . . . . 5 (∃𝑧((𝑥𝐵𝑧𝑧𝐴𝑦) ∨ (𝑥𝐶𝑧𝑧𝐴𝑦)) ↔ (∃𝑧(𝑥𝐵𝑧𝑧𝐴𝑦) ∨ ∃𝑧(𝑥𝐶𝑧𝑧𝐴𝑦)))
86, 7bitr2i 277 . . . 4 ((∃𝑧(𝑥𝐵𝑧𝑧𝐴𝑦) ∨ ∃𝑧(𝑥𝐶𝑧𝑧𝐴𝑦)) ↔ ∃𝑧(𝑥(𝐵𝐶)𝑧𝑧𝐴𝑦))
98opabbii 5035 . . 3 {⟨𝑥, 𝑦⟩ ∣ (∃𝑧(𝑥𝐵𝑧𝑧𝐴𝑦) ∨ ∃𝑧(𝑥𝐶𝑧𝑧𝐴𝑦))} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥(𝐵𝐶)𝑧𝑧𝐴𝑦)}
101, 9eqtri 2821 . 2 ({⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐵𝑧𝑧𝐴𝑦)} ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐶𝑧𝑧𝐴𝑦)}) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥(𝐵𝐶)𝑧𝑧𝐴𝑦)}
11 df-co 5459 . . 3 (𝐴𝐵) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐵𝑧𝑧𝐴𝑦)}
12 df-co 5459 . . 3 (𝐴𝐶) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐶𝑧𝑧𝐴𝑦)}
1311, 12uneq12i 4064 . 2 ((𝐴𝐵) ∪ (𝐴𝐶)) = ({⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐵𝑧𝑧𝐴𝑦)} ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐶𝑧𝑧𝐴𝑦)})
14 df-co 5459 . 2 (𝐴 ∘ (𝐵𝐶)) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥(𝐵𝐶)𝑧𝑧𝐴𝑦)}
1510, 13, 143eqtr4ri 2832 1 (𝐴 ∘ (𝐵𝐶)) = ((𝐴𝐵) ∪ (𝐴𝐶))
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 396   ∨ wo 842   = wceq 1525  ∃wex 1765   ∪ cun 3863   class class class wbr 4968  {copab 5030   ∘ ccom 5454 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-ext 2771 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-v 3442  df-un 3870  df-br 4969  df-opab 5031  df-co 5459 This theorem is referenced by:  mvdco  18308  ustssco  22510  coprprop  30119  cycpmconjv  30417  cvmliftlem10  32151  poimirlem9  34453  diophren  38916  rtrclex  39483  trclubgNEW  39484  trclexi  39486  rtrclexi  39487  cnvtrcl0  39492  trrelsuperrel2dg  39522  cotrclrcl  39593  frege131d  39615
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