Theorem coundir 5041

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 4007	. . 3 ⊢ ({⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐴𝑧)} ∪ {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐵𝑧)}) = {⟨𝑥, 𝑧⟩ ∣ (∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐴𝑧) ∨ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐵𝑧))}
2		brun 3979	. . . . . . . 8 ⊢ (𝑦(𝐴 ∪ 𝐵)𝑧 ↔ (𝑦𝐴𝑧 ∨ 𝑦𝐵𝑧))
3	2	anbi2i 452	. . . . . . 7 ⊢ ((𝑥𝐶𝑦 ∧ 𝑦(𝐴 ∪ 𝐵)𝑧) ↔ (𝑥𝐶𝑦 ∧ (𝑦𝐴𝑧 ∨ 𝑦𝐵𝑧)))
4		andi 807	. . . . . . 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 1584	. . . . 5 ⊢ (∃𝑦(𝑥𝐶𝑦 ∧ 𝑦(𝐴 ∪ 𝐵)𝑧) ↔ ∃𝑦((𝑥𝐶𝑦 ∧ 𝑦𝐴𝑧) ∨ (𝑥𝐶𝑦 ∧ 𝑦𝐵𝑧)))
7		19.43 1607	. . . . 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 3995	. . 3 ⊢ {⟨𝑥, 𝑧⟩ ∣ (∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐴𝑧) ∨ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐵𝑧))} = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦(𝐴 ∪ 𝐵)𝑧)}
10	1, 9	eqtri 2160	. 2 ⊢ ({⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐴𝑧)} ∪ {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐵𝑧)}) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦(𝐴 ∪ 𝐵)𝑧)}
11		df-co 4548	. . 3 ⊢ (𝐴 ∘ 𝐶) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐴𝑧)}
12		df-co 4548	. . 3 ⊢ (𝐵 ∘ 𝐶) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐵𝑧)}
13	11, 12	uneq12i 3228	. 2 ⊢ ((𝐴 ∘ 𝐶) ∪ (𝐵 ∘ 𝐶)) = ({⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐴𝑧)} ∪ {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦𝐵𝑧)})
14		df-co 4548	. 2 ⊢ ((𝐴 ∪ 𝐵) ∘ 𝐶) = {⟨𝑥, 𝑧⟩ ∣ ∃𝑦(𝑥𝐶𝑦 ∧ 𝑦(𝐴 ∪ 𝐵)𝑧)}
15	10, 13, 14	3eqtr4ri 2171	1 ⊢ ((𝐴 ∪ 𝐵) ∘ 𝐶) = ((𝐴 ∘ 𝐶) ∪ (𝐵 ∘ 𝐶))

Syntax hints:   ∧ wa 103   ∨ wo 697   = wceq 1331  ∃wex 1468   ∪ cun 3069   class class class wbr 3929  {copab 3988   ∘ ccom 4543

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-un 3075  df-br 3930  df-opab 3990  df-co 4548

This theorem is referenced by: (None)