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Theorem coundir 5798
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 4880 . . 3 ({⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦𝑦𝐴𝑧)} ∪ {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦𝑦𝐵𝑧)}) = {⟨𝑥, 𝑧⟩ ∣ (∃𝑦(𝑥𝐶𝑦𝑦𝐴𝑧) ∨ ∃𝑦(𝑥𝐶𝑦𝑦𝐵𝑧))}
2 brun 4855 . . . . . . . 8 (𝑦(𝐴𝐵)𝑧 ↔ (𝑦𝐴𝑧𝑦𝐵𝑧))
32anbi2i 732 . . . . . . 7 ((𝑥𝐶𝑦𝑦(𝐴𝐵)𝑧) ↔ (𝑥𝐶𝑦 ∧ (𝑦𝐴𝑧𝑦𝐵𝑧)))
4 andi 947 . . . . . . 7 ((𝑥𝐶𝑦 ∧ (𝑦𝐴𝑧𝑦𝐵𝑧)) ↔ ((𝑥𝐶𝑦𝑦𝐴𝑧) ∨ (𝑥𝐶𝑦𝑦𝐵𝑧)))
53, 4bitri 264 . . . . . 6 ((𝑥𝐶𝑦𝑦(𝐴𝐵)𝑧) ↔ ((𝑥𝐶𝑦𝑦𝐴𝑧) ∨ (𝑥𝐶𝑦𝑦𝐵𝑧)))
65exbii 1923 . . . . 5 (∃𝑦(𝑥𝐶𝑦𝑦(𝐴𝐵)𝑧) ↔ ∃𝑦((𝑥𝐶𝑦𝑦𝐴𝑧) ∨ (𝑥𝐶𝑦𝑦𝐵𝑧)))
7 19.43 1959 . . . . 5 (∃𝑦((𝑥𝐶𝑦𝑦𝐴𝑧) ∨ (𝑥𝐶𝑦𝑦𝐵𝑧)) ↔ (∃𝑦(𝑥𝐶𝑦𝑦𝐴𝑧) ∨ ∃𝑦(𝑥𝐶𝑦𝑦𝐵𝑧)))
86, 7bitr2i 265 . . . 4 ((∃𝑦(𝑥𝐶𝑦𝑦𝐴𝑧) ∨ ∃𝑦(𝑥𝐶𝑦𝑦𝐵𝑧)) ↔ ∃𝑦(𝑥𝐶𝑦𝑦(𝐴𝐵)𝑧))
98opabbii 4869 . . 3 {⟨𝑥, 𝑧⟩ ∣ (∃𝑦(𝑥𝐶𝑦𝑦𝐴𝑧) ∨ ∃𝑦(𝑥𝐶𝑦𝑦𝐵𝑧))} = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦𝑦(𝐴𝐵)𝑧)}
101, 9eqtri 2782 . 2 ({⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦𝑦𝐴𝑧)} ∪ {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦𝑦𝐵𝑧)}) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦𝑦(𝐴𝐵)𝑧)}
11 df-co 5275 . . 3 (𝐴𝐶) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦𝑦𝐴𝑧)}
12 df-co 5275 . . 3 (𝐵𝐶) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦𝑦𝐵𝑧)}
1311, 12uneq12i 3908 . 2 ((𝐴𝐶) ∪ (𝐵𝐶)) = ({⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦𝑦𝐴𝑧)} ∪ {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦𝑦𝐵𝑧)})
14 df-co 5275 . 2 ((𝐴𝐵) ∘ 𝐶) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦𝑦(𝐴𝐵)𝑧)}
1510, 13, 143eqtr4ri 2793 1 ((𝐴𝐵) ∘ 𝐶) = ((𝐴𝐶) ∪ (𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  wo 382  wa 383   = wceq 1632  wex 1853  cun 3713   class class class wbr 4804  {copab 4864  ccom 5270
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-v 3342  df-un 3720  df-br 4805  df-opab 4865  df-co 5275
This theorem is referenced by:  diophrw  37824  diophren  37879  rtrclex  38426  trclubgNEW  38427  trclexi  38429  rtrclexi  38430  cnvtrcl0  38435  trrelsuperrel2dg  38465
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