Theorem coundi 4765

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 3827	. . 3 ⊢ ({⟨x, y⟩ ∣ ∃z(xBz ∧ zAy)} ∪ {⟨x, y⟩ ∣ ∃z(x𝐶z ∧ zAy)}) = {⟨x, y⟩ ∣ (∃z(xBz ∧ zAy) ∨ ∃z(x𝐶z ∧ zAy))}
2		brun 3801	. . . . . . . 8 ⊢ (x(B ∪ 𝐶)z ↔ (xBz ∨ x𝐶z))
3	2	anbi1i 431	. . . . . . 7 ⊢ ((x(B ∪ 𝐶)z ∧ zAy) ↔ ((xBz ∨ x𝐶z) ∧ zAy))
4		andir 731	. . . . . . 7 ⊢ (((xBz ∨ x𝐶z) ∧ zAy) ↔ ((xBz ∧ zAy) ∨ (x𝐶z ∧ zAy)))
5	3, 4	bitri 173	. . . . . 6 ⊢ ((x(B ∪ 𝐶)z ∧ zAy) ↔ ((xBz ∧ zAy) ∨ (x𝐶z ∧ zAy)))
6	5	exbii 1493	. . . . 5 ⊢ (∃z(x(B ∪ 𝐶)z ∧ zAy) ↔ ∃z((xBz ∧ zAy) ∨ (x𝐶z ∧ zAy)))
7		19.43 1516	. . . . 5 ⊢ (∃z((xBz ∧ zAy) ∨ (x𝐶z ∧ zAy)) ↔ (∃z(xBz ∧ zAy) ∨ ∃z(x𝐶z ∧ zAy)))
8	6, 7	bitr2i 174	. . . 4 ⊢ ((∃z(xBz ∧ zAy) ∨ ∃z(x𝐶z ∧ zAy)) ↔ ∃z(x(B ∪ 𝐶)z ∧ zAy))
9	8	opabbii 3815	. . 3 ⊢ {⟨x, y⟩ ∣ (∃z(xBz ∧ zAy) ∨ ∃z(x𝐶z ∧ zAy))} = {⟨x, y⟩ ∣ ∃z(x(B ∪ 𝐶)z ∧ zAy)}
10	1, 9	eqtri 2057	. 2 ⊢ ({⟨x, y⟩ ∣ ∃z(xBz ∧ zAy)} ∪ {⟨x, y⟩ ∣ ∃z(x𝐶z ∧ zAy)}) = {⟨x, y⟩ ∣ ∃z(x(B ∪ 𝐶)z ∧ zAy)}
11		df-co 4297	. . 3 ⊢ (A ∘ B) = {⟨x, y⟩ ∣ ∃z(xBz ∧ zAy)}
12		df-co 4297	. . 3 ⊢ (A ∘ 𝐶) = {⟨x, y⟩ ∣ ∃z(x𝐶z ∧ zAy)}
13	11, 12	uneq12i 3089	. 2 ⊢ ((A ∘ B) ∪ (A ∘ 𝐶)) = ({⟨x, y⟩ ∣ ∃z(xBz ∧ zAy)} ∪ {⟨x, y⟩ ∣ ∃z(x𝐶z ∧ zAy)})
14		df-co 4297	. 2 ⊢ (A ∘ (B ∪ 𝐶)) = {⟨x, y⟩ ∣ ∃z(x(B ∪ 𝐶)z ∧ zAy)}
15	10, 13, 14	3eqtr4ri 2068	1 ⊢ (A ∘ (B ∪ 𝐶)) = ((A ∘ B) ∪ (A ∘ 𝐶))

Syntax hints:   ∧ wa 97   ∨ wo 628   = wceq 1242  ∃wex 1378   ∪ cun 2909   class class class wbr 3755  {copab 3808   ∘ ccom 4292

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-br 3756  df-opab 3810  df-co 4297

This theorem is referenced by:  relcoi1  4792