Theorem coundi 6205

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 5166	. . 3 ⊢ ({⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)} ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐶𝑧 ∧ 𝑧𝐴𝑦)}) = {⟨𝑥, 𝑦⟩ ∣ (∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ∨ ∃𝑧(𝑥𝐶𝑧 ∧ 𝑧𝐴𝑦))}
2		brun 5137	. . . . . . . 8 ⊢ (𝑥(𝐵 ∪ 𝐶)𝑧 ↔ (𝑥𝐵𝑧 ∨ 𝑥𝐶𝑧))
3	2	anbi1i 625	. . . . . . 7 ⊢ ((𝑥(𝐵 ∪ 𝐶)𝑧 ∧ 𝑧𝐴𝑦) ↔ ((𝑥𝐵𝑧 ∨ 𝑥𝐶𝑧) ∧ 𝑧𝐴𝑦))
4		andir 1011	. . . . . . 7 ⊢ (((𝑥𝐵𝑧 ∨ 𝑥𝐶𝑧) ∧ 𝑧𝐴𝑦) ↔ ((𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ∨ (𝑥𝐶𝑧 ∧ 𝑧𝐴𝑦)))
5	3, 4	bitri 275	. . . . . 6 ⊢ ((𝑥(𝐵 ∪ 𝐶)𝑧 ∧ 𝑧𝐴𝑦) ↔ ((𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ∨ (𝑥𝐶𝑧 ∧ 𝑧𝐴𝑦)))
6	5	exbii 1850	. . . . 5 ⊢ (∃𝑧(𝑥(𝐵 ∪ 𝐶)𝑧 ∧ 𝑧𝐴𝑦) ↔ ∃𝑧((𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ∨ (𝑥𝐶𝑧 ∧ 𝑧𝐴𝑦)))
7		19.43 1884	. . . . 5 ⊢ (∃𝑧((𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ∨ (𝑥𝐶𝑧 ∧ 𝑧𝐴𝑦)) ↔ (∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ∨ ∃𝑧(𝑥𝐶𝑧 ∧ 𝑧𝐴𝑦)))
8	6, 7	bitr2i 276	. . . 4 ⊢ ((∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ∨ ∃𝑧(𝑥𝐶𝑧 ∧ 𝑧𝐴𝑦)) ↔ ∃𝑧(𝑥(𝐵 ∪ 𝐶)𝑧 ∧ 𝑧𝐴𝑦))
9	8	opabbii 5153	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ (∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ∨ ∃𝑧(𝑥𝐶𝑧 ∧ 𝑧𝐴𝑦))} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥(𝐵 ∪ 𝐶)𝑧 ∧ 𝑧𝐴𝑦)}
10	1, 9	eqtri 2760	. 2 ⊢ ({⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)} ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐶𝑧 ∧ 𝑧𝐴𝑦)}) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥(𝐵 ∪ 𝐶)𝑧 ∧ 𝑧𝐴𝑦)}
11		df-co 5633	. . 3 ⊢ (𝐴 ∘ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)}
12		df-co 5633	. . 3 ⊢ (𝐴 ∘ 𝐶) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐶𝑧 ∧ 𝑧𝐴𝑦)}
13	11, 12	uneq12i 4107	. 2 ⊢ ((𝐴 ∘ 𝐵) ∪ (𝐴 ∘ 𝐶)) = ({⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)} ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐶𝑧 ∧ 𝑧𝐴𝑦)})
14		df-co 5633	. 2 ⊢ (𝐴 ∘ (𝐵 ∪ 𝐶)) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥(𝐵 ∪ 𝐶)𝑧 ∧ 𝑧𝐴𝑦)}
15	10, 13, 14	3eqtr4ri 2771	1 ⊢ (𝐴 ∘ (𝐵 ∪ 𝐶)) = ((𝐴 ∘ 𝐵) ∪ (𝐴 ∘ 𝐶))

Syntax hints:   ∧ wa 395   ∨ wo 848   = wceq 1542  ∃wex 1781   ∪ cun 3888   class class class wbr 5086  {copab 5148   ∘ ccom 5628

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3432  df-un 3895  df-br 5087  df-opab 5149  df-co 5633

This theorem is referenced by:  f1ofvswap  7254  mvdco  19411  ustssco  24190  coprprop  32787  cycpmconjv  33218  cvmliftlem10  35492  poimirlem9  37964  diophren  43259  rtrclex  44062  trclubgNEW  44063  trclexi  44065  rtrclexi  44066  cnvtrcl0  44071  trrelsuperrel2dg  44116  cotrclrcl  44187  frege131d  44209  dftpos6  49362