Theorem coundi 4745

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 3806	. . 3 ⊢ ({⟨x, y⟩ ∣ ∃z(xBz ∧ zAy)} ∪ {⟨x, y⟩ ∣ ∃z(x𝐶z ∧ zAy)}) = {⟨x, y⟩ ∣ (∃z(xBz ∧ zAy) ∨ ∃z(x𝐶z ∧ zAy))}
2		brun 3780	. . . . . . . 8 ⊢ (x(B ∪ 𝐶)z ↔ (xBz ∨ x𝐶z))
3	2	anbi1i 434	. . . . . . 7 ⊢ ((x(B ∪ 𝐶)z ∧ zAy) ↔ ((xBz ∨ x𝐶z) ∧ zAy))
4		andir 720	. . . . . . 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 1474	. . . . 5 ⊢ (∃z(x(B ∪ 𝐶)z ∧ zAy) ↔ ∃z((xBz ∧ zAy) ∨ (x𝐶z ∧ zAy)))
7		19.43 1497	. . . . 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 3794	. . 3 ⊢ {⟨x, y⟩ ∣ (∃z(xBz ∧ zAy) ∨ ∃z(x𝐶z ∧ zAy))} = {⟨x, y⟩ ∣ ∃z(x(B ∪ 𝐶)z ∧ zAy)}
10	1, 9	eqtri 2038	. 2 ⊢ ({⟨x, y⟩ ∣ ∃z(xBz ∧ zAy)} ∪ {⟨x, y⟩ ∣ ∃z(x𝐶z ∧ zAy)}) = {⟨x, y⟩ ∣ ∃z(x(B ∪ 𝐶)z ∧ zAy)}
11		df-co 4277	. . 3 ⊢ (A ∘ B) = {⟨x, y⟩ ∣ ∃z(xBz ∧ zAy)}
12		df-co 4277	. . 3 ⊢ (A ∘ 𝐶) = {⟨x, y⟩ ∣ ∃z(x𝐶z ∧ zAy)}
13	11, 12	uneq12i 3068	. 2 ⊢ ((A ∘ B) ∪ (A ∘ 𝐶)) = ({⟨x, y⟩ ∣ ∃z(xBz ∧ zAy)} ∪ {⟨x, y⟩ ∣ ∃z(x𝐶z ∧ zAy)})
14		df-co 4277	. 2 ⊢ (A ∘ (B ∪ 𝐶)) = {⟨x, y⟩ ∣ ∃z(x(B ∪ 𝐶)z ∧ zAy)}
15	10, 13, 14	3eqtr4ri 2049	1 ⊢ (A ∘ (B ∪ 𝐶)) = ((A ∘ B) ∪ (A ∘ 𝐶))

Syntax hints:   ∧ wa 97   ∨ wo 616   = wceq 1226  ∃wex 1358   ∪ cun 2888   class class class wbr 3734  {copab 3787   ∘ ccom 4272

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 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000

This theorem depends on definitions:  df-bi 110  df-tru 1229  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-v 2533  df-un 2895  df-br 3735  df-opab 3789  df-co 4277

This theorem is referenced by:  relcoi1  4772