Theorem coundir 5049

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 4015	. . 3 ⊢ ({⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐴𝑧)} ∪ {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐵𝑧)}) = {⟨𝑥, 𝑧⟩ ∣ (∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐴𝑧) ∨ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐵𝑧))}
2		brun 3987	. . . . . . . 8 ⊢ (𝑦(𝐴 ∪ 𝐵)𝑧 ↔ (𝑦𝐴𝑧 ∨ 𝑦𝐵𝑧))
3	2	anbi2i 453	. . . . . . 7 ⊢ ((𝑥𝐶𝑦 ∧ 𝑦(𝐴 ∪ 𝐵)𝑧) ↔ (𝑥𝐶𝑦 ∧ (𝑦𝐴𝑧 ∨ 𝑦𝐵𝑧)))
4		andi 808	. . . . . . 7 ⊢ ((𝑥𝐶𝑦 ∧ (𝑦𝐴𝑧 ∨ 𝑦𝐵𝑧)) ↔ ((𝑥𝐶𝑦 ∧ 𝑦𝐴𝑧) ∨ (𝑥𝐶𝑦 ∧ 𝑦𝐵𝑧)))
5	3, 4	bitri 183	. . . . . 6 ⊢ ((𝑥𝐶𝑦 ∧ 𝑦(𝐴 ∪ 𝐵)𝑧) ↔ ((𝑥𝐶𝑦 ∧ 𝑦𝐴𝑧) ∨ (𝑥𝐶𝑦 ∧ 𝑦𝐵𝑧)))
6	5	exbii 1585	. . . . 5 ⊢ (∃𝑦(𝑥𝐶𝑦 ∧ 𝑦(𝐴 ∪ 𝐵)𝑧) ↔ ∃𝑦((𝑥𝐶𝑦 ∧ 𝑦𝐴𝑧) ∨ (𝑥𝐶𝑦 ∧ 𝑦𝐵𝑧)))
7		19.43 1608	. . . . 5 ⊢ (∃𝑦((𝑥𝐶𝑦 ∧ 𝑦𝐴𝑧) ∨ (𝑥𝐶𝑦 ∧ 𝑦𝐵𝑧)) ↔ (∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐴𝑧) ∨ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐵𝑧)))
8	6, 7	bitr2i 184	. . . 4 ⊢ ((∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐴𝑧) ∨ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐵𝑧)) ↔ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦(𝐴 ∪ 𝐵)𝑧))
9	8	opabbii 4003	. . 3 ⊢ {⟨𝑥, 𝑧⟩ ∣ (∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐴𝑧) ∨ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐵𝑧))} = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦(𝐴 ∪ 𝐵)𝑧)}
10	1, 9	eqtri 2161	. 2 ⊢ ({⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐴𝑧)} ∪ {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐵𝑧)}) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦(𝐴 ∪ 𝐵)𝑧)}
11		df-co 4556	. . 3 ⊢ (𝐴 ∘ 𝐶) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐴𝑧)}
12		df-co 4556	. . 3 ⊢ (𝐵 ∘ 𝐶) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐵𝑧)}
13	11, 12	uneq12i 3233	. 2 ⊢ ((𝐴 ∘ 𝐶) ∪ (𝐵 ∘ 𝐶)) = ({⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐴𝑧)} ∪ {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐵𝑧)})
14		df-co 4556	. 2 ⊢ ((𝐴 ∪ 𝐵) ∘ 𝐶) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦(𝐴 ∪ 𝐵)𝑧)}
15	10, 13, 14	3eqtr4ri 2172	1 ⊢ ((𝐴 ∪ 𝐵) ∘ 𝐶) = ((𝐴 ∘ 𝐶) ∪ (𝐵 ∘ 𝐶))

Syntax hints:   ∧ wa 103   ∨ wo 698   = wceq 1332  ∃wex 1469   ∪ cun 3074   class class class wbr 3937  {copab 3996   ∘ ccom 4551

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-un 3080  df-br 3938  df-opab 3998  df-co 4556

This theorem is referenced by: (None)