Theorem coundi 6238

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 ∘ (𝐵 ∪ 𝐶)) = ((𝐴 ∘ 𝐵) ∪ (𝐴 ∘ 𝐶))

Proof of Theorem coundi

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		unopab 5185	. . 3 ⊢ ({⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)} ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐶𝑧 ∧ 𝑧𝐴𝑦)}) = {⟨𝑥, 𝑦⟩ ∣ (∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ∨ ∃𝑧(𝑥𝐶𝑧 ∧ 𝑧𝐴𝑦))}
2		brun 5156	. . . . . . . 8 ⊢ (𝑥(𝐵 ∪ 𝐶)𝑧 ↔ (𝑥𝐵𝑧 ∨ 𝑥𝐶𝑧))
3	2	anbi1i 635	. . . . . . 7 ⊢ ((𝑥(𝐵 ∪ 𝐶)𝑧 ∧ 𝑧𝐴𝑦) ↔ ((𝑥𝐵𝑧 ∨ 𝑥𝐶𝑧) ∧ 𝑧𝐴𝑦))
4		andir 1024	. . . . . . 7 ⊢ (((𝑥𝐵𝑧 ∨ 𝑥𝐶𝑧) ∧ 𝑧𝐴𝑦) ↔ ((𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ∨ (𝑥𝐶𝑧 ∧ 𝑧𝐴𝑦)))
5	3, 4	bitri 278	. . . . . 6 ⊢ ((𝑥(𝐵 ∪ 𝐶)𝑧 ∧ 𝑧𝐴𝑦) ↔ ((𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ∨ (𝑥𝐶𝑧 ∧ 𝑧𝐴𝑦)))
6	5	exbii 1871	. . . . 5 ⊢ (∃𝑧(𝑥(𝐵 ∪ 𝐶)𝑧 ∧ 𝑧𝐴𝑦) ↔ ∃𝑧((𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ∨ (𝑥𝐶𝑧 ∧ 𝑧𝐴𝑦)))
7		19.43 1905	. . . . 5 ⊢ (∃𝑧((𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ∨ (𝑥𝐶𝑧 ∧ 𝑧𝐴𝑦)) ↔ (∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ∨ ∃𝑧(𝑥𝐶𝑧 ∧ 𝑧𝐴𝑦)))
8	6, 7	bitr2i 279	. . . 4 ⊢ ((∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ∨ ∃𝑧(𝑥𝐶𝑧 ∧ 𝑧𝐴𝑦)) ↔ ∃𝑧(𝑥(𝐵 ∪ 𝐶)𝑧 ∧ 𝑧𝐴𝑦))
9	8	opabbii 5172	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ (∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ∨ ∃𝑧(𝑥𝐶𝑧 ∧ 𝑧𝐴𝑦))} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥(𝐵 ∪ 𝐶)𝑧 ∧ 𝑧𝐴𝑦)}
10	1, 9	eqtri 2788	. 2 ⊢ ({⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)} ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐶𝑧 ∧ 𝑧𝐴𝑦)}) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥(𝐵 ∪ 𝐶)𝑧 ∧ 𝑧𝐴𝑦)}
11		df-co 5661	. . 3 ⊢ (𝐴 ∘ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)}
12		df-co 5661	. . 3 ⊢ (𝐴 ∘ 𝐶) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐶𝑧 ∧ 𝑧𝐴𝑦)}
13	11, 12	uneq12i 4122	. 2 ⊢ ((𝐴 ∘ 𝐵) ∪ (𝐴 ∘ 𝐶)) = ({⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)} ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐶𝑧 ∧ 𝑧𝐴𝑦)})
14		df-co 5661	. 2 ⊢ (𝐴 ∘ (𝐵 ∪ 𝐶)) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥(𝐵 ∪ 𝐶)𝑧 ∧ 𝑧𝐴𝑦)}
15	10, 13, 14	3eqtr4ri 2799	1 ⊢ (𝐴 ∘ (𝐵 ∪ 𝐶)) = ((𝐴 ∘ 𝐵) ∪ (𝐴 ∘ 𝐶))

Syntax hints:   ∧ wa 400   ∨ wo 860   = wceq 1563  ∃wex 1802   ∪ cun 3905   class class class wbr 5105  {copab 5167   ∘ ccom 5656

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1566  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-v 3459  df-un 3912  df-br 5106  df-opab 5168  df-co 5661

This theorem is referenced by:  f1ofvswap  7294  mvdco  19506  ustssco  24333  coprprop  32956  cycpmconjv  33375  cvmliftlem10  35657  poimirlem9  38140  diophren  43402  rtrclex  44205  trclubgNEW  44206  trclexi  44208  rtrclexi  44209  cnvtrcl0  44214  trrelsuperrel2dg  44259  cotrclrcl  44330  frege131d  44352  dftpos6  49504