Theorem coiun 6213

Description: Composition with an indexed union. (Contributed by NM, 21-Dec-2008.)

Assertion

Ref	Expression
coiun	⊢ (𝐴 ∘ ∪ 𝑥 ∈ 𝐶 𝐵) = ∪ 𝑥 ∈ 𝐶 (𝐴 ∘ 𝐵)

Distinct variable group:   𝑥,𝐴

Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥)

Proof of Theorem coiun

Dummy variables 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relco 6065	. 2 ⊢ Rel (𝐴 ∘ ∪ 𝑥 ∈ 𝐶 𝐵)
2		reliun 5763	. . 3 ⊢ (Rel ∪ 𝑥 ∈ 𝐶 (𝐴 ∘ 𝐵) ↔ ∀𝑥 ∈ 𝐶 Rel (𝐴 ∘ 𝐵))
3		relco 6065	. . . 4 ⊢ Rel (𝐴 ∘ 𝐵)
4	3	a1i 11	. . 3 ⊢ (𝑥 ∈ 𝐶 → Rel (𝐴 ∘ 𝐵))
5	2, 4	mprgbir 3056	. 2 ⊢ Rel ∪ 𝑥 ∈ 𝐶 (𝐴 ∘ 𝐵)
6		eliun 4948	. . . . . . . . 9 ⊢ (⟨𝑦, 𝑤⟩ ∈ ∪ 𝑥 ∈ 𝐶 𝐵 ↔ ∃𝑥 ∈ 𝐶 ⟨𝑦, 𝑤⟩ ∈ 𝐵)
7		df-br 5097	. . . . . . . . 9 ⊢ (𝑦∪ 𝑥 ∈ 𝐶 𝐵𝑤 ↔ ⟨𝑦, 𝑤⟩ ∈ ∪ 𝑥 ∈ 𝐶 𝐵)
8		df-br 5097	. . . . . . . . . 10 ⊢ (𝑦𝐵𝑤 ↔ ⟨𝑦, 𝑤⟩ ∈ 𝐵)
9	8	rexbii 3081	. . . . . . . . 9 ⊢ (∃𝑥 ∈ 𝐶 𝑦𝐵𝑤 ↔ ∃𝑥 ∈ 𝐶 ⟨𝑦, 𝑤⟩ ∈ 𝐵)
10	6, 7, 9	3bitr4i 303	. . . . . . . 8 ⊢ (𝑦∪ 𝑥 ∈ 𝐶 𝐵𝑤 ↔ ∃𝑥 ∈ 𝐶 𝑦𝐵𝑤)
11	10	anbi1i 624	. . . . . . 7 ⊢ ((𝑦∪ 𝑥 ∈ 𝐶 𝐵𝑤 ∧ 𝑤𝐴𝑧) ↔ (∃𝑥 ∈ 𝐶 𝑦𝐵𝑤 ∧ 𝑤𝐴𝑧))
12		r19.41v 3164	. . . . . . 7 ⊢ (∃𝑥 ∈ 𝐶 (𝑦𝐵𝑤 ∧ 𝑤𝐴𝑧) ↔ (∃𝑥 ∈ 𝐶 𝑦𝐵𝑤 ∧ 𝑤𝐴𝑧))
13	11, 12	bitr4i 278	. . . . . 6 ⊢ ((𝑦∪ 𝑥 ∈ 𝐶 𝐵𝑤 ∧ 𝑤𝐴𝑧) ↔ ∃𝑥 ∈ 𝐶 (𝑦𝐵𝑤 ∧ 𝑤𝐴𝑧))
14	13	exbii 1849	. . . . 5 ⊢ (∃𝑤(𝑦∪ 𝑥 ∈ 𝐶 𝐵𝑤 ∧ 𝑤𝐴𝑧) ↔ ∃𝑤∃𝑥 ∈ 𝐶 (𝑦𝐵𝑤 ∧ 𝑤𝐴𝑧))
15		rexcom4 3261	. . . . 5 ⊢ (∃𝑥 ∈ 𝐶 ∃𝑤(𝑦𝐵𝑤 ∧ 𝑤𝐴𝑧) ↔ ∃𝑤∃𝑥 ∈ 𝐶 (𝑦𝐵𝑤 ∧ 𝑤𝐴𝑧))
16	14, 15	bitr4i 278	. . . 4 ⊢ (∃𝑤(𝑦∪ 𝑥 ∈ 𝐶 𝐵𝑤 ∧ 𝑤𝐴𝑧) ↔ ∃𝑥 ∈ 𝐶 ∃𝑤(𝑦𝐵𝑤 ∧ 𝑤𝐴𝑧))
17		vex 3442	. . . . 5 ⊢ 𝑦 ∈ V
18		vex 3442	. . . . 5 ⊢ 𝑧 ∈ V
19	17, 18	opelco 5818	. . . 4 ⊢ (⟨𝑦, 𝑧⟩ ∈ (𝐴 ∘ ∪ 𝑥 ∈ 𝐶 𝐵) ↔ ∃𝑤(𝑦∪ 𝑥 ∈ 𝐶 𝐵𝑤 ∧ 𝑤𝐴𝑧))
20	17, 18	opelco 5818	. . . . 5 ⊢ (⟨𝑦, 𝑧⟩ ∈ (𝐴 ∘ 𝐵) ↔ ∃𝑤(𝑦𝐵𝑤 ∧ 𝑤𝐴𝑧))
21	20	rexbii 3081	. . . 4 ⊢ (∃𝑥 ∈ 𝐶 ⟨𝑦, 𝑧⟩ ∈ (𝐴 ∘ 𝐵) ↔ ∃𝑥 ∈ 𝐶 ∃𝑤(𝑦𝐵𝑤 ∧ 𝑤𝐴𝑧))
22	16, 19, 21	3bitr4i 303	. . 3 ⊢ (⟨𝑦, 𝑧⟩ ∈ (𝐴 ∘ ∪ 𝑥 ∈ 𝐶 𝐵) ↔ ∃𝑥 ∈ 𝐶 ⟨𝑦, 𝑧⟩ ∈ (𝐴 ∘ 𝐵))
23		eliun 4948	. . 3 ⊢ (⟨𝑦, 𝑧⟩ ∈ ∪ 𝑥 ∈ 𝐶 (𝐴 ∘ 𝐵) ↔ ∃𝑥 ∈ 𝐶 ⟨𝑦, 𝑧⟩ ∈ (𝐴 ∘ 𝐵))
24	22, 23	bitr4i 278	. 2 ⊢ (⟨𝑦, 𝑧⟩ ∈ (𝐴 ∘ ∪ 𝑥 ∈ 𝐶 𝐵) ↔ ⟨𝑦, 𝑧⟩ ∈ ∪ 𝑥 ∈ 𝐶 (𝐴 ∘ 𝐵))
25	1, 5, 24	eqrelriiv 5737	1 ⊢ (𝐴 ∘ ∪ 𝑥 ∈ 𝐶 𝐵) = ∪ 𝑥 ∈ 𝐶 (𝐴 ∘ 𝐵)

Syntax hints:   ∧ wa 395   = wceq 1541  ∃wex 1780   ∈ wcel 2113  ∃wrex 3058  ⟨cop 4584  ∪ ciun 4944   class class class wbr 5096   ∘ ccom 5626  Rel wrel 5627

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-11 2162  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4284  df-if 4478  df-sn 4579  df-pr 4581  df-op 4585  df-iun 4946  df-br 5097  df-opab 5159  df-xp 5628  df-rel 5629  df-co 5631

This theorem is referenced by:  fparlem3  8054  fparlem4  8055  trclrelexplem  43894  trclfvcom  43906