Theorem coundir 6242

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 5205	. . 3 ⊢ ({⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐴𝑧)} ∪ {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐵𝑧)}) = {⟨𝑥, 𝑧⟩ ∣ (∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐴𝑧) ∨ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐵𝑧))}
2		brun 5175	. . . . . . . 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 5191	. . 3 ⊢ {⟨𝑥, 𝑧⟩ ∣ (∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐴𝑧) ∨ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐵𝑧))} = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦(𝐴 ∪ 𝐵)𝑧)}
10	1, 9	eqtri 2759	. 2 ⊢ ({⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐴𝑧)} ∪ {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐵𝑧)}) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦(𝐴 ∪ 𝐵)𝑧)}
11		df-co 5668	. . 3 ⊢ (𝐴 ∘ 𝐶) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐴𝑧)}
12		df-co 5668	. . 3 ⊢ (𝐵 ∘ 𝐶) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐵𝑧)}
13	11, 12	uneq12i 4146	. 2 ⊢ ((𝐴 ∘ 𝐶) ∪ (𝐵 ∘ 𝐶)) = ({⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐴𝑧)} ∪ {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐵𝑧)})
14		df-co 5668	. 2 ⊢ ((𝐴 ∪ 𝐵) ∘ 𝐶) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦(𝐴 ∪ 𝐵)𝑧)}
15	10, 13, 14	3eqtr4ri 2770	1 ⊢ ((𝐴 ∪ 𝐵) ∘ 𝐶) = ((𝐴 ∘ 𝐶) ∪ (𝐵 ∘ 𝐶))

Syntax hints:   ∧ wa 395   ∨ wo 847   = wceq 1540  ∃wex 1779   ∪ cun 3929   class class class wbr 5124  {copab 5186   ∘ ccom 5663

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 2008  ax-8 2111  ax-9 2119  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-v 3466  df-un 3936  df-br 5125  df-opab 5187  df-co 5668

This theorem is referenced by:  coprprop  32681  cycpmconjv  33158  diophrw  42757  diophren  42811  rtrclex  43616  trclubgNEW  43617  trclexi  43619  rtrclexi  43620  cnvtrcl0  43625  trrelsuperrel2dg  43670