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Theorem coundir 6248
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 5231 . . 3 ({⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦𝑦𝐴𝑧)} ∪ {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦𝑦𝐵𝑧)}) = {⟨𝑥, 𝑧⟩ ∣ (∃𝑦(𝑥𝐶𝑦𝑦𝐴𝑧) ∨ ∃𝑦(𝑥𝐶𝑦𝑦𝐵𝑧))}
2 brun 5200 . . . . . . . 8 (𝑦(𝐴𝐵)𝑧 ↔ (𝑦𝐴𝑧𝑦𝐵𝑧))
32anbi2i 624 . . . . . . 7 ((𝑥𝐶𝑦𝑦(𝐴𝐵)𝑧) ↔ (𝑥𝐶𝑦 ∧ (𝑦𝐴𝑧𝑦𝐵𝑧)))
4 andi 1007 . . . . . . 7 ((𝑥𝐶𝑦 ∧ (𝑦𝐴𝑧𝑦𝐵𝑧)) ↔ ((𝑥𝐶𝑦𝑦𝐴𝑧) ∨ (𝑥𝐶𝑦𝑦𝐵𝑧)))
53, 4bitri 275 . . . . . 6 ((𝑥𝐶𝑦𝑦(𝐴𝐵)𝑧) ↔ ((𝑥𝐶𝑦𝑦𝐴𝑧) ∨ (𝑥𝐶𝑦𝑦𝐵𝑧)))
65exbii 1851 . . . . 5 (∃𝑦(𝑥𝐶𝑦𝑦(𝐴𝐵)𝑧) ↔ ∃𝑦((𝑥𝐶𝑦𝑦𝐴𝑧) ∨ (𝑥𝐶𝑦𝑦𝐵𝑧)))
7 19.43 1886 . . . . 5 (∃𝑦((𝑥𝐶𝑦𝑦𝐴𝑧) ∨ (𝑥𝐶𝑦𝑦𝐵𝑧)) ↔ (∃𝑦(𝑥𝐶𝑦𝑦𝐴𝑧) ∨ ∃𝑦(𝑥𝐶𝑦𝑦𝐵𝑧)))
86, 7bitr2i 276 . . . 4 ((∃𝑦(𝑥𝐶𝑦𝑦𝐴𝑧) ∨ ∃𝑦(𝑥𝐶𝑦𝑦𝐵𝑧)) ↔ ∃𝑦(𝑥𝐶𝑦𝑦(𝐴𝐵)𝑧))
98opabbii 5216 . . 3 {⟨𝑥, 𝑧⟩ ∣ (∃𝑦(𝑥𝐶𝑦𝑦𝐴𝑧) ∨ ∃𝑦(𝑥𝐶𝑦𝑦𝐵𝑧))} = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦𝑦(𝐴𝐵)𝑧)}
101, 9eqtri 2761 . 2 ({⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦𝑦𝐴𝑧)} ∪ {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦𝑦𝐵𝑧)}) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦𝑦(𝐴𝐵)𝑧)}
11 df-co 5686 . . 3 (𝐴𝐶) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦𝑦𝐴𝑧)}
12 df-co 5686 . . 3 (𝐵𝐶) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦𝑦𝐵𝑧)}
1311, 12uneq12i 4162 . 2 ((𝐴𝐶) ∪ (𝐵𝐶)) = ({⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦𝑦𝐴𝑧)} ∪ {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦𝑦𝐵𝑧)})
14 df-co 5686 . 2 ((𝐴𝐵) ∘ 𝐶) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦𝑦(𝐴𝐵)𝑧)}
1510, 13, 143eqtr4ri 2772 1 ((𝐴𝐵) ∘ 𝐶) = ((𝐴𝐶) ∪ (𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  wa 397  wo 846   = wceq 1542  wex 1782  cun 3947   class class class wbr 5149  {copab 5211  ccom 5681
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-un 3954  df-br 5150  df-opab 5212  df-co 5686
This theorem is referenced by:  coprprop  31921  cycpmconjv  32301  diophrw  41497  diophren  41551  rtrclex  42368  trclubgNEW  42369  trclexi  42371  rtrclexi  42372  cnvtrcl0  42377  trrelsuperrel2dg  42422
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