Theorem coundir 5084

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
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 4639	. . 3 ⊢ ({⟨x, z⟩ ∣ ∃y(xCy ∧ yAz)} ∪ {⟨x, z⟩ ∣ ∃y(xCy ∧ yBz)}) = {⟨x, z⟩ ∣ (∃y(xCy ∧ yAz) ∨ ∃y(xCy ∧ yBz))}
2		brun 4693	. . . . . . . 8 ⊢ (y(A ∪ B)z ↔ (yAz ∨ yBz))
3	2	anbi2i 675	. . . . . . 7 ⊢ ((xCy ∧ y(A ∪ B)z) ↔ (xCy ∧ (yAz ∨ yBz)))
4		andi 837	. . . . . . 7 ⊢ ((xCy ∧ (yAz ∨ yBz)) ↔ ((xCy ∧ yAz) ∨ (xCy ∧ yBz)))
5	3, 4	bitri 240	. . . . . 6 ⊢ ((xCy ∧ y(A ∪ B)z) ↔ ((xCy ∧ yAz) ∨ (xCy ∧ yBz)))
6	5	exbii 1582	. . . . 5 ⊢ (∃y(xCy ∧ y(A ∪ B)z) ↔ ∃y((xCy ∧ yAz) ∨ (xCy ∧ yBz)))
7		19.43 1605	. . . . 5 ⊢ (∃y((xCy ∧ yAz) ∨ (xCy ∧ yBz)) ↔ (∃y(xCy ∧ yAz) ∨ ∃y(xCy ∧ yBz)))
8	6, 7	bitr2i 241	. . . 4 ⊢ ((∃y(xCy ∧ yAz) ∨ ∃y(xCy ∧ yBz)) ↔ ∃y(xCy ∧ y(A ∪ B)z))
9	8	opabbii 4627	. . 3 ⊢ {⟨x, z⟩ ∣ (∃y(xCy ∧ yAz) ∨ ∃y(xCy ∧ yBz))} = {⟨x, z⟩ ∣ ∃y(xCy ∧ y(A ∪ B)z)}
10	1, 9	eqtri 2373	. 2 ⊢ ({⟨x, z⟩ ∣ ∃y(xCy ∧ yAz)} ∪ {⟨x, z⟩ ∣ ∃y(xCy ∧ yBz)}) = {⟨x, z⟩ ∣ ∃y(xCy ∧ y(A ∪ B)z)}
11		df-co 4727	. . 3 ⊢ (A ∘ C) = {⟨x, z⟩ ∣ ∃y(xCy ∧ yAz)}
12		df-co 4727	. . 3 ⊢ (B ∘ C) = {⟨x, z⟩ ∣ ∃y(xCy ∧ yBz)}
13	11, 12	uneq12i 3417	. 2 ⊢ ((A ∘ C) ∪ (B ∘ C)) = ({⟨x, z⟩ ∣ ∃y(xCy ∧ yAz)} ∪ {⟨x, z⟩ ∣ ∃y(xCy ∧ yBz)})
14		df-co 4727	. 2 ⊢ ((A ∪ B) ∘ C) = {⟨x, z⟩ ∣ ∃y(xCy ∧ y(A ∪ B)z)}
15	10, 13, 14	3eqtr4ri 2384	1 ⊢ ((A ∪ B) ∘ C) = ((A ∘ C) ∪ (B ∘ C))

Syntax hints:   ∨ wo 357   ∧ wa 358  ∃wex 1541   = wceq 1642   ∪ cun 3208  {copab 4623   class class class wbr 4640   ∘ ccom 4722

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 2479  df-v 2862  df-nin 3212  df-compl 3213  df-un 3215  df-opab 4624  df-br 4641  df-co 4727

This theorem is referenced by: (None)