Theorem coundir 6249

Description: Class composition distributes over union. (Contributed by NM, 21-Dec-2008.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Assertion

Ref	Expression
coundir	⊢ ((𝐴 ∪ 𝐵) ∘ 𝐶) = ((𝐴 ∘ 𝐶) ∪ (𝐵 ∘ 𝐶))

Proof of Theorem coundir

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

Step	Hyp	Ref	Expression
1		unopab 5184	. . 3 ⊢ ({⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐴𝑧)} ∪ {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐵𝑧)}) = {⟨𝑥, 𝑧⟩ ∣ (∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐴𝑧) ∨ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐵𝑧))}
2		brun 5155	. . . . . . . 8 ⊢ (𝑦(𝐴 ∪ 𝐵)𝑧 ↔ (𝑦𝐴𝑧 ∨ 𝑦𝐵𝑧))
3	2	anbi2i 635	. . . . . . 7 ⊢ ((𝑥𝐶𝑦 ∧ 𝑦(𝐴 ∪ 𝐵)𝑧) ↔ (𝑥𝐶𝑦 ∧ (𝑦𝐴𝑧 ∨ 𝑦𝐵𝑧)))
4		andi 1025	. . . . . . 7 ⊢ ((𝑥𝐶𝑦 ∧ (𝑦𝐴𝑧 ∨ 𝑦𝐵𝑧)) ↔ ((𝑥𝐶𝑦 ∧ 𝑦𝐴𝑧) ∨ (𝑥𝐶𝑦 ∧ 𝑦𝐵𝑧)))
5	3, 4	bitri 278	. . . . . 6 ⊢ ((𝑥𝐶𝑦 ∧ 𝑦(𝐴 ∪ 𝐵)𝑧) ↔ ((𝑥𝐶𝑦 ∧ 𝑦𝐴𝑧) ∨ (𝑥𝐶𝑦 ∧ 𝑦𝐵𝑧)))
6	5	exbii 1880	. . . . 5 ⊢ (∃𝑦(𝑥𝐶𝑦 ∧ 𝑦(𝐴 ∪ 𝐵)𝑧) ↔ ∃𝑦((𝑥𝐶𝑦 ∧ 𝑦𝐴𝑧) ∨ (𝑥𝐶𝑦 ∧ 𝑦𝐵𝑧)))
7		19.43 1914	. . . . 5 ⊢ (∃𝑦((𝑥𝐶𝑦 ∧ 𝑦𝐴𝑧) ∨ (𝑥𝐶𝑦 ∧ 𝑦𝐵𝑧)) ↔ (∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐴𝑧) ∨ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐵𝑧)))
8	6, 7	bitr2i 279	. . . 4 ⊢ ((∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐴𝑧) ∨ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐵𝑧)) ↔ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦(𝐴 ∪ 𝐵)𝑧))
9	8	opabbii 5171	. . 3 ⊢ {⟨𝑥, 𝑧⟩ ∣ (∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐴𝑧) ∨ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐵𝑧))} = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦(𝐴 ∪ 𝐵)𝑧)}
10	1, 9	eqtri 2783	. 2 ⊢ ({⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐴𝑧)} ∪ {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐵𝑧)}) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦(𝐴 ∪ 𝐵)𝑧)}
11		df-co 5659	. . 3 ⊢ (𝐴 ∘ 𝐶) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐴𝑧)}
12		df-co 5659	. . 3 ⊢ (𝐵 ∘ 𝐶) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐵𝑧)}
13	11, 12	uneq12i 4165	. 2 ⊢ ((𝐴 ∘ 𝐶) ∪ (𝐵 ∘ 𝐶)) = ({⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐴𝑧)} ∪ {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐵𝑧)})
14		df-co 5659	. 2 ⊢ ((𝐴 ∪ 𝐵) ∘ 𝐶) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦(𝐴 ∪ 𝐵)𝑧)}
15	10, 13, 14	3eqtr4ri 2794	1 ⊢ ((𝐴 ∪ 𝐵) ∘ 𝐶) = ((𝐴 ∘ 𝐶) ∪ (𝐵 ∘ 𝐶))

This proof depends on syntax axioms:   ∧ wa 401   ∨ wo 861   = wceq 1569  ∃wex 1811   ∪ cun 4149   class class class wbr 5102  {copab 5166   ∘ ccom 5654

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-un 4151  df-br 5103  df-opab 5167  df-co 5659

This theorem is used by:  coprprop  33339  cycpmconjv  33749  diophrw  43863  diophren  43913  rtrclex  44710  trclubgNEW  44711  trclexi  44713  rtrclexi  44714  cnvtrcl0  44719  trrelsuperrel2dg  44764