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Theorem coundir 5781
Description: Class composition distributes over union. (Contributed by NM, 21-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
coundir ((𝐴𝐵) ∘ 𝐶) = ((𝐴𝐶) ∪ (𝐵𝐶))

Proof of Theorem coundir
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 unopab 4862 . . 3 ({⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦𝑦𝐴𝑧)} ∪ {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦𝑦𝐵𝑧)}) = {⟨𝑥, 𝑧⟩ ∣ (∃𝑦(𝑥𝐶𝑦𝑦𝐴𝑧) ∨ ∃𝑦(𝑥𝐶𝑦𝑦𝐵𝑧))}
2 brun 4837 . . . . . . . 8 (𝑦(𝐴𝐵)𝑧 ↔ (𝑦𝐴𝑧𝑦𝐵𝑧))
32anbi2i 609 . . . . . . 7 ((𝑥𝐶𝑦𝑦(𝐴𝐵)𝑧) ↔ (𝑥𝐶𝑦 ∧ (𝑦𝐴𝑧𝑦𝐵𝑧)))
4 andi 987 . . . . . . 7 ((𝑥𝐶𝑦 ∧ (𝑦𝐴𝑧𝑦𝐵𝑧)) ↔ ((𝑥𝐶𝑦𝑦𝐴𝑧) ∨ (𝑥𝐶𝑦𝑦𝐵𝑧)))
53, 4bitri 264 . . . . . 6 ((𝑥𝐶𝑦𝑦(𝐴𝐵)𝑧) ↔ ((𝑥𝐶𝑦𝑦𝐴𝑧) ∨ (𝑥𝐶𝑦𝑦𝐵𝑧)))
65exbii 1924 . . . . 5 (∃𝑦(𝑥𝐶𝑦𝑦(𝐴𝐵)𝑧) ↔ ∃𝑦((𝑥𝐶𝑦𝑦𝐴𝑧) ∨ (𝑥𝐶𝑦𝑦𝐵𝑧)))
7 19.43 1962 . . . . 5 (∃𝑦((𝑥𝐶𝑦𝑦𝐴𝑧) ∨ (𝑥𝐶𝑦𝑦𝐵𝑧)) ↔ (∃𝑦(𝑥𝐶𝑦𝑦𝐴𝑧) ∨ ∃𝑦(𝑥𝐶𝑦𝑦𝐵𝑧)))
86, 7bitr2i 265 . . . 4 ((∃𝑦(𝑥𝐶𝑦𝑦𝐴𝑧) ∨ ∃𝑦(𝑥𝐶𝑦𝑦𝐵𝑧)) ↔ ∃𝑦(𝑥𝐶𝑦𝑦(𝐴𝐵)𝑧))
98opabbii 4851 . . 3 {⟨𝑥, 𝑧⟩ ∣ (∃𝑦(𝑥𝐶𝑦𝑦𝐴𝑧) ∨ ∃𝑦(𝑥𝐶𝑦𝑦𝐵𝑧))} = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦𝑦(𝐴𝐵)𝑧)}
101, 9eqtri 2793 . 2 ({⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦𝑦𝐴𝑧)} ∪ {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦𝑦𝐵𝑧)}) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦𝑦(𝐴𝐵)𝑧)}
11 df-co 5258 . . 3 (𝐴𝐶) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦𝑦𝐴𝑧)}
12 df-co 5258 . . 3 (𝐵𝐶) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦𝑦𝐵𝑧)}
1311, 12uneq12i 3916 . 2 ((𝐴𝐶) ∪ (𝐵𝐶)) = ({⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦𝑦𝐴𝑧)} ∪ {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦𝑦𝐵𝑧)})
14 df-co 5258 . 2 ((𝐴𝐵) ∘ 𝐶) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦𝑦(𝐴𝐵)𝑧)}
1510, 13, 143eqtr4ri 2804 1 ((𝐴𝐵) ∘ 𝐶) = ((𝐴𝐶) ∪ (𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  wa 382  wo 834   = wceq 1631  wex 1852  cun 3721   class class class wbr 4786  {copab 4846  ccom 5253
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-v 3353  df-un 3728  df-br 4787  df-opab 4847  df-co 5258
This theorem is referenced by:  diophrw  37848  diophren  37903  rtrclex  38450  trclubgNEW  38451  trclexi  38453  rtrclexi  38454  cnvtrcl0  38459  trrelsuperrel2dg  38489
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