Theorem coundi 5289

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 4210	. . 3 ⊢ ({⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)} ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐶𝑧 ∧ 𝑧𝐴𝑦)}) = {⟨𝑥, 𝑦⟩ ∣ (∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ∨ ∃𝑧(𝑥𝐶𝑧 ∧ 𝑧𝐴𝑦))}
2		brun 4182	. . . . . . . 8 ⊢ (𝑥(𝐵 ∪ 𝐶)𝑧 ↔ (𝑥𝐵𝑧 ∨ 𝑥𝐶𝑧))
3	2	anbi1i 462	. . . . . . 7 ⊢ ((𝑥(𝐵 ∪ 𝐶)𝑧 ∧ 𝑧𝐴𝑦) ↔ ((𝑥𝐵𝑧 ∨ 𝑥𝐶𝑧) ∧ 𝑧𝐴𝑦))
4		andir 831	. . . . . . 7 ⊢ (((𝑥𝐵𝑧 ∨ 𝑥𝐶𝑧) ∧ 𝑧𝐴𝑦) ↔ ((𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ∨ (𝑥𝐶𝑧 ∧ 𝑧𝐴𝑦)))
5	3, 4	bitri 184	. . . . . 6 ⊢ ((𝑥(𝐵 ∪ 𝐶)𝑧 ∧ 𝑧𝐴𝑦) ↔ ((𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ∨ (𝑥𝐶𝑧 ∧ 𝑧𝐴𝑦)))
6	5	exbii 1658	. . . . 5 ⊢ (∃𝑧(𝑥(𝐵 ∪ 𝐶)𝑧 ∧ 𝑧𝐴𝑦) ↔ ∃𝑧((𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ∨ (𝑥𝐶𝑧 ∧ 𝑧𝐴𝑦)))
7		19.43 1681	. . . . 5 ⊢ (∃𝑧((𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ∨ (𝑥𝐶𝑧 ∧ 𝑧𝐴𝑦)) ↔ (∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ∨ ∃𝑧(𝑥𝐶𝑧 ∧ 𝑧𝐴𝑦)))
8	6, 7	bitr2i 185	. . . 4 ⊢ ((∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ∨ ∃𝑧(𝑥𝐶𝑧 ∧ 𝑧𝐴𝑦)) ↔ ∃𝑧(𝑥(𝐵 ∪ 𝐶)𝑧 ∧ 𝑧𝐴𝑦))
9	8	opabbii 4198	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ (∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦) ∨ ∃𝑧(𝑥𝐶𝑧 ∧ 𝑧𝐴𝑦))} = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥(𝐵 ∪ 𝐶)𝑧 ∧ 𝑧𝐴𝑦)}
10	1, 9	eqtri 2259	. 2 ⊢ ({⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)} ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐶𝑧 ∧ 𝑧𝐴𝑦)}) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥(𝐵 ∪ 𝐶)𝑧 ∧ 𝑧𝐴𝑦)}
11		df-co 4783	. . 3 ⊢ (𝐴 ∘ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)}
12		df-co 4783	. . 3 ⊢ (𝐴 ∘ 𝐶) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐶𝑧 ∧ 𝑧𝐴𝑦)}
13	11, 12	uneq12i 3381	. 2 ⊢ ((𝐴 ∘ 𝐵) ∪ (𝐴 ∘ 𝐶)) = ({⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐵𝑧 ∧ 𝑧𝐴𝑦)} ∪ {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥𝐶𝑧 ∧ 𝑧𝐴𝑦)})
14		df-co 4783	. 2 ⊢ (𝐴 ∘ (𝐵 ∪ 𝐶)) = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧(𝑥(𝐵 ∪ 𝐶)𝑧 ∧ 𝑧𝐴𝑦)}
15	10, 13, 14	3eqtr4ri 2270	1 ⊢ (𝐴 ∘ (𝐵 ∪ 𝐶)) = ((𝐴 ∘ 𝐵) ∪ (𝐴 ∘ 𝐶))

This proof depends on syntax axioms:   ∧ wa 104   ∨ wo 720   = wceq 1402  ∃wex 1545   ∪ cun 3218   class class class wbr 4130  {copab 4191   ∘ ccom 4778

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-br 4131  df-opab 4193  df-co 4783

This theorem is used by:  relcoi1  5319