Theorem coundi 5082

Description: Class composition distributes over union. (The proof was shortened by Andrew Salmon, 27-Aug-2011.) (Contributed by set.mm contributors, 21-Dec-2008.) (Revised by set.mm contributors, 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 4638	. . 3 ⊢ ({⟨x, y⟩ ∣ ∃z(xBz ∧ zAy)} ∪ {⟨x, y⟩ ∣ ∃z(xCz ∧ zAy)}) = {⟨x, y⟩ ∣ (∃z(xBz ∧ zAy) ∨ ∃z(xCz ∧ zAy))}
2		brun 4692	. . . . . . . 8 ⊢ (x(B ∪ C)z ↔ (xBz ∨ xCz))
3	2	anbi1i 676	. . . . . . 7 ⊢ ((x(B ∪ C)z ∧ zAy) ↔ ((xBz ∨ xCz) ∧ zAy))
4		andir 838	. . . . . . 7 ⊢ (((xBz ∨ xCz) ∧ zAy) ↔ ((xBz ∧ zAy) ∨ (xCz ∧ zAy)))
5	3, 4	bitri 240	. . . . . 6 ⊢ ((x(B ∪ C)z ∧ zAy) ↔ ((xBz ∧ zAy) ∨ (xCz ∧ zAy)))
6	5	exbii 1582	. . . . 5 ⊢ (∃z(x(B ∪ C)z ∧ zAy) ↔ ∃z((xBz ∧ zAy) ∨ (xCz ∧ zAy)))
7		19.43 1605	. . . . 5 ⊢ (∃z((xBz ∧ zAy) ∨ (xCz ∧ zAy)) ↔ (∃z(xBz ∧ zAy) ∨ ∃z(xCz ∧ zAy)))
8	6, 7	bitr2i 241	. . . 4 ⊢ ((∃z(xBz ∧ zAy) ∨ ∃z(xCz ∧ zAy)) ↔ ∃z(x(B ∪ C)z ∧ zAy))
9	8	opabbii 4626	. . 3 ⊢ {⟨x, y⟩ ∣ (∃z(xBz ∧ zAy) ∨ ∃z(xCz ∧ zAy))} = {⟨x, y⟩ ∣ ∃z(x(B ∪ C)z ∧ zAy)}
10	1, 9	eqtri 2373	. 2 ⊢ ({⟨x, y⟩ ∣ ∃z(xBz ∧ zAy)} ∪ {⟨x, y⟩ ∣ ∃z(xCz ∧ zAy)}) = {⟨x, y⟩ ∣ ∃z(x(B ∪ C)z ∧ zAy)}
11		df-co 4726	. . 3 ⊢ (A ∘ B) = {⟨x, y⟩ ∣ ∃z(xBz ∧ zAy)}
12		df-co 4726	. . 3 ⊢ (A ∘ C) = {⟨x, y⟩ ∣ ∃z(xCz ∧ zAy)}
13	11, 12	uneq12i 3416	. 2 ⊢ ((A ∘ B) ∪ (A ∘ C)) = ({⟨x, y⟩ ∣ ∃z(xBz ∧ zAy)} ∪ {⟨x, y⟩ ∣ ∃z(xCz ∧ zAy)})
14		df-co 4726	. 2 ⊢ (A ∘ (B ∪ C)) = {⟨x, y⟩ ∣ ∃z(x(B ∪ C)z ∧ zAy)}
15	10, 13, 14	3eqtr4ri 2384	1 ⊢ (A ∘ (B ∪ C)) = ((A ∘ B) ∪ (A ∘ C))

Syntax hints:   ∨ wo 357   ∧ wa 358  ∃wex 1541   = wceq 1642   ∪ cun 3207  {copab 4622   class class class wbr 4639   ∘ ccom 4721

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-un 3214  df-opab 4623  df-br 4640  df-co 4726

This theorem is referenced by: (None)