Theorem coiun 6245

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 6095	. 2 ⊢ Rel (𝐴 ∘ ∪ 𝑥 ∈ 𝐶 𝐵)
2		reliun 5795	. . 3 ⊢ (Rel ∪ 𝑥 ∈ 𝐶 (𝐴 ∘ 𝐵) ↔ ∀𝑥 ∈ 𝐶 Rel (𝐴 ∘ 𝐵))
3		relco 6095	. . . 4 ⊢ Rel (𝐴 ∘ 𝐵)
4	3	a1i 11	. . 3 ⊢ (𝑥 ∈ 𝐶 → Rel (𝐴 ∘ 𝐵))
5	2, 4	mprgbir 3058	. 2 ⊢ Rel ∪ 𝑥 ∈ 𝐶 (𝐴 ∘ 𝐵)
6		eliun 4971	. . . . . . . . 9 ⊢ (⟨𝑦, 𝑤⟩ ∈ ∪ 𝑥 ∈ 𝐶 𝐵 ↔ ∃𝑥 ∈ 𝐶 ⟨𝑦, 𝑤⟩ ∈ 𝐵)
7		df-br 5120	. . . . . . . . 9 ⊢ (𝑦∪ 𝑥 ∈ 𝐶 𝐵𝑤 ↔ ⟨𝑦, 𝑤⟩ ∈ ∪ 𝑥 ∈ 𝐶 𝐵)
8		df-br 5120	. . . . . . . . . 10 ⊢ (𝑦𝐵𝑤 ↔ ⟨𝑦, 𝑤⟩ ∈ 𝐵)
9	8	rexbii 3083	. . . . . . . . 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 3174	. . . . . . 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 1848	. . . . 5 ⊢ (∃𝑤(𝑦∪ 𝑥 ∈ 𝐶 𝐵𝑤 ∧ 𝑤𝐴𝑧) ↔ ∃𝑤∃𝑥 ∈ 𝐶 (𝑦𝐵𝑤 ∧ 𝑤𝐴𝑧))
15		rexcom4 3269	. . . . 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 3463	. . . . 5 ⊢ 𝑦 ∈ V
18		vex 3463	. . . . 5 ⊢ 𝑧 ∈ V
19	17, 18	opelco 5851	. . . 4 ⊢ (⟨𝑦, 𝑧⟩ ∈ (𝐴 ∘ ∪ 𝑥 ∈ 𝐶 𝐵) ↔ ∃𝑤(𝑦∪ 𝑥 ∈ 𝐶 𝐵𝑤 ∧ 𝑤𝐴𝑧))
20	17, 18	opelco 5851	. . . . 5 ⊢ (⟨𝑦, 𝑧⟩ ∈ (𝐴 ∘ 𝐵) ↔ ∃𝑤(𝑦𝐵𝑤 ∧ 𝑤𝐴𝑧))
21	20	rexbii 3083	. . . 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 4971	. . 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 5769	1 ⊢ (𝐴 ∘ ∪ 𝑥 ∈ 𝐶 𝐵) = ∪ 𝑥 ∈ 𝐶 (𝐴 ∘ 𝐵)

Syntax hints:   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2108  ∃wrex 3060  ⟨cop 4607  ∪ ciun 4967   class class class wbr 5119   ∘ ccom 5658  Rel wrel 5659

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-iun 4969  df-br 5120  df-opab 5182  df-xp 5660  df-rel 5661  df-co 5663

This theorem is referenced by:  fparlem3  8113  fparlem4  8114  trclrelexplem  43735  trclfvcom  43747