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Theorem coundi 4919
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 3909 . . 3 ({⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐵𝑧𝑧𝐴𝑦)} ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐶𝑧𝑧𝐴𝑦)}) = {⟨𝑥, 𝑦⟩ ∣ (∃𝑧(𝑥𝐵𝑧𝑧𝐴𝑦) ∨ ∃𝑧(𝑥𝐶𝑧𝑧𝐴𝑦))}
2 brun 3883 . . . . . . . 8 (𝑥(𝐵𝐶)𝑧 ↔ (𝑥𝐵𝑧𝑥𝐶𝑧))
32anbi1i 446 . . . . . . 7 ((𝑥(𝐵𝐶)𝑧𝑧𝐴𝑦) ↔ ((𝑥𝐵𝑧𝑥𝐶𝑧) ∧ 𝑧𝐴𝑦))
4 andir 768 . . . . . . 7 (((𝑥𝐵𝑧𝑥𝐶𝑧) ∧ 𝑧𝐴𝑦) ↔ ((𝑥𝐵𝑧𝑧𝐴𝑦) ∨ (𝑥𝐶𝑧𝑧𝐴𝑦)))
53, 4bitri 182 . . . . . 6 ((𝑥(𝐵𝐶)𝑧𝑧𝐴𝑦) ↔ ((𝑥𝐵𝑧𝑧𝐴𝑦) ∨ (𝑥𝐶𝑧𝑧𝐴𝑦)))
65exbii 1541 . . . . 5 (∃𝑧(𝑥(𝐵𝐶)𝑧𝑧𝐴𝑦) ↔ ∃𝑧((𝑥𝐵𝑧𝑧𝐴𝑦) ∨ (𝑥𝐶𝑧𝑧𝐴𝑦)))
7 19.43 1564 . . . . 5 (∃𝑧((𝑥𝐵𝑧𝑧𝐴𝑦) ∨ (𝑥𝐶𝑧𝑧𝐴𝑦)) ↔ (∃𝑧(𝑥𝐵𝑧𝑧𝐴𝑦) ∨ ∃𝑧(𝑥𝐶𝑧𝑧𝐴𝑦)))
86, 7bitr2i 183 . . . 4 ((∃𝑧(𝑥𝐵𝑧𝑧𝐴𝑦) ∨ ∃𝑧(𝑥𝐶𝑧𝑧𝐴𝑦)) ↔ ∃𝑧(𝑥(𝐵𝐶)𝑧𝑧𝐴𝑦))
98opabbii 3897 . . 3 {⟨𝑥, 𝑦⟩ ∣ (∃𝑧(𝑥𝐵𝑧𝑧𝐴𝑦) ∨ ∃𝑧(𝑥𝐶𝑧𝑧𝐴𝑦))} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥(𝐵𝐶)𝑧𝑧𝐴𝑦)}
101, 9eqtri 2108 . 2 ({⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐵𝑧𝑧𝐴𝑦)} ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐶𝑧𝑧𝐴𝑦)}) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥(𝐵𝐶)𝑧𝑧𝐴𝑦)}
11 df-co 4437 . . 3 (𝐴𝐵) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐵𝑧𝑧𝐴𝑦)}
12 df-co 4437 . . 3 (𝐴𝐶) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐶𝑧𝑧𝐴𝑦)}
1311, 12uneq12i 3150 . 2 ((𝐴𝐵) ∪ (𝐴𝐶)) = ({⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐵𝑧𝑧𝐴𝑦)} ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐶𝑧𝑧𝐴𝑦)})
14 df-co 4437 . 2 (𝐴 ∘ (𝐵𝐶)) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥(𝐵𝐶)𝑧𝑧𝐴𝑦)}
1510, 13, 143eqtr4ri 2119 1 (𝐴 ∘ (𝐵𝐶)) = ((𝐴𝐵) ∪ (𝐴𝐶))
Colors of variables: wff set class
Syntax hints:  wa 102  wo 664   = wceq 1289  wex 1426  cun 2995   class class class wbr 3837  {copab 3890  ccom 4432
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3001  df-br 3838  df-opab 3892  df-co 4437
This theorem is referenced by:  relcoi1  4949
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