Theorem coundir 6237

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 5200	. . 3 ⊢ ({⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐴𝑧)} ∪ {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐵𝑧)}) = {⟨𝑥, 𝑧⟩ ∣ (∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐴𝑧) ∨ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐵𝑧))}
2		brun 5170	. . . . . . . 8 ⊢ (𝑦(𝐴 ∪ 𝐵)𝑧 ↔ (𝑦𝐴𝑧 ∨ 𝑦𝐵𝑧))
3	2	anbi2i 623	. . . . . . 7 ⊢ ((𝑥𝐶𝑦 ∧ 𝑦(𝐴 ∪ 𝐵)𝑧) ↔ (𝑥𝐶𝑦 ∧ (𝑦𝐴𝑧 ∨ 𝑦𝐵𝑧)))
4		andi 1009	. . . . . . 7 ⊢ ((𝑥𝐶𝑦 ∧ (𝑦𝐴𝑧 ∨ 𝑦𝐵𝑧)) ↔ ((𝑥𝐶𝑦 ∧ 𝑦𝐴𝑧) ∨ (𝑥𝐶𝑦 ∧ 𝑦𝐵𝑧)))
5	3, 4	bitri 275	. . . . . 6 ⊢ ((𝑥𝐶𝑦 ∧ 𝑦(𝐴 ∪ 𝐵)𝑧) ↔ ((𝑥𝐶𝑦 ∧ 𝑦𝐴𝑧) ∨ (𝑥𝐶𝑦 ∧ 𝑦𝐵𝑧)))
6	5	exbii 1848	. . . . 5 ⊢ (∃𝑦(𝑥𝐶𝑦 ∧ 𝑦(𝐴 ∪ 𝐵)𝑧) ↔ ∃𝑦((𝑥𝐶𝑦 ∧ 𝑦𝐴𝑧) ∨ (𝑥𝐶𝑦 ∧ 𝑦𝐵𝑧)))
7		19.43 1882	. . . . 5 ⊢ (∃𝑦((𝑥𝐶𝑦 ∧ 𝑦𝐴𝑧) ∨ (𝑥𝐶𝑦 ∧ 𝑦𝐵𝑧)) ↔ (∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐴𝑧) ∨ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐵𝑧)))
8	6, 7	bitr2i 276	. . . 4 ⊢ ((∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐴𝑧) ∨ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐵𝑧)) ↔ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦(𝐴 ∪ 𝐵)𝑧))
9	8	opabbii 5186	. . 3 ⊢ {⟨𝑥, 𝑧⟩ ∣ (∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐴𝑧) ∨ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐵𝑧))} = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦(𝐴 ∪ 𝐵)𝑧)}
10	1, 9	eqtri 2758	. 2 ⊢ ({⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐴𝑧)} ∪ {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐵𝑧)}) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦(𝐴 ∪ 𝐵)𝑧)}
11		df-co 5663	. . 3 ⊢ (𝐴 ∘ 𝐶) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐴𝑧)}
12		df-co 5663	. . 3 ⊢ (𝐵 ∘ 𝐶) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐵𝑧)}
13	11, 12	uneq12i 4141	. 2 ⊢ ((𝐴 ∘ 𝐶) ∪ (𝐵 ∘ 𝐶)) = ({⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐴𝑧)} ∪ {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐵𝑧)})
14		df-co 5663	. 2 ⊢ ((𝐴 ∪ 𝐵) ∘ 𝐶) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦(𝐴 ∪ 𝐵)𝑧)}
15	10, 13, 14	3eqtr4ri 2769	1 ⊢ ((𝐴 ∪ 𝐵) ∘ 𝐶) = ((𝐴 ∘ 𝐶) ∪ (𝐵 ∘ 𝐶))

Syntax hints:   ∧ wa 395   ∨ wo 847   = wceq 1540  ∃wex 1779   ∪ cun 3924   class class class wbr 5119  {copab 5181   ∘ ccom 5658

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-v 3461  df-un 3931  df-br 5120  df-opab 5182  df-co 5663

This theorem is referenced by:  coprprop  32622  cycpmconjv  33099  diophrw  42729  diophren  42783  rtrclex  43588  trclubgNEW  43589  trclexi  43591  rtrclexi  43592  cnvtrcl0  43597  trrelsuperrel2dg  43642