Theorem coiun 5253

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 5242	. 2 ⊢ Rel (𝐴 ∘ ∪ 𝑥 ∈ 𝐶 𝐵)
2		reliun 4854	. . 3 ⊢ (Rel ∪ 𝑥 ∈ 𝐶 (𝐴 ∘ 𝐵) ↔ ∀𝑥 ∈ 𝐶 Rel (𝐴 ∘ 𝐵))
3		relco 5242	. . . 4 ⊢ Rel (𝐴 ∘ 𝐵)
4	3	a1i 9	. . 3 ⊢ (𝑥 ∈ 𝐶 → Rel (𝐴 ∘ 𝐵))
5	2, 4	mprgbir 2591	. 2 ⊢ Rel ∪ 𝑥 ∈ 𝐶 (𝐴 ∘ 𝐵)
6		eliun 3979	. . . . . . . 8 ⊢ (⟨𝑦, 𝑤⟩ ∈ ∪ 𝑥 ∈ 𝐶 𝐵 ↔ ∃𝑥 ∈ 𝐶 ⟨𝑦, 𝑤⟩ ∈ 𝐵)
7		df-br 4094	. . . . . . . 8 ⊢ (𝑦∪ 𝑥 ∈ 𝐶 𝐵𝑤 ↔ ⟨𝑦, 𝑤⟩ ∈ ∪ 𝑥 ∈ 𝐶 𝐵)
8		df-br 4094	. . . . . . . . 9 ⊢ (𝑦𝐵𝑤 ↔ ⟨𝑦, 𝑤⟩ ∈ 𝐵)
9	8	rexbii 2540	. . . . . . . 8 ⊢ (∃𝑥 ∈ 𝐶 𝑦𝐵𝑤 ↔ ∃𝑥 ∈ 𝐶 ⟨𝑦, 𝑤⟩ ∈ 𝐵)
10	6, 7, 9	3bitr4i 212	. . . . . . 7 ⊢ (𝑦∪ 𝑥 ∈ 𝐶 𝐵𝑤 ↔ ∃𝑥 ∈ 𝐶 𝑦𝐵𝑤)
11	10	anbi1i 458	. . . . . 6 ⊢ ((𝑦∪ 𝑥 ∈ 𝐶 𝐵𝑤 ∧ 𝑤𝐴𝑧) ↔ (∃𝑥 ∈ 𝐶 𝑦𝐵𝑤 ∧ 𝑤𝐴𝑧))
12		r19.41v 2690	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐶 (𝑦𝐵𝑤 ∧ 𝑤𝐴𝑧) ↔ (∃𝑥 ∈ 𝐶 𝑦𝐵𝑤 ∧ 𝑤𝐴𝑧))
13	11, 12	bitr4i 187	. . . . 5 ⊢ ((𝑦∪ 𝑥 ∈ 𝐶 𝐵𝑤 ∧ 𝑤𝐴𝑧) ↔ ∃𝑥 ∈ 𝐶 (𝑦𝐵𝑤 ∧ 𝑤𝐴𝑧))
14	13	exbii 1654	. . . 4 ⊢ (∃𝑤(𝑦∪ 𝑥 ∈ 𝐶 𝐵𝑤 ∧ 𝑤𝐴𝑧) ↔ ∃𝑤∃𝑥 ∈ 𝐶 (𝑦𝐵𝑤 ∧ 𝑤𝐴𝑧))
15		rexcom4 2827	. . . 4 ⊢ (∃𝑥 ∈ 𝐶 ∃𝑤(𝑦𝐵𝑤 ∧ 𝑤𝐴𝑧) ↔ ∃𝑤∃𝑥 ∈ 𝐶 (𝑦𝐵𝑤 ∧ 𝑤𝐴𝑧))
16	14, 15	bitr4i 187	. . 3 ⊢ (∃𝑤(𝑦∪ 𝑥 ∈ 𝐶 𝐵𝑤 ∧ 𝑤𝐴𝑧) ↔ ∃𝑥 ∈ 𝐶 ∃𝑤(𝑦𝐵𝑤 ∧ 𝑤𝐴𝑧))
17		vex 2806	. . . 4 ⊢ 𝑦 ∈ V
18		vex 2806	. . . 4 ⊢ 𝑧 ∈ V
19	17, 18	opelco 4908	. . 3 ⊢ (⟨𝑦, 𝑧⟩ ∈ (𝐴 ∘ ∪ 𝑥 ∈ 𝐶 𝐵) ↔ ∃𝑤(𝑦∪ 𝑥 ∈ 𝐶 𝐵𝑤 ∧ 𝑤𝐴𝑧))
20		eliun 3979	. . . 4 ⊢ (⟨𝑦, 𝑧⟩ ∈ ∪ 𝑥 ∈ 𝐶 (𝐴 ∘ 𝐵) ↔ ∃𝑥 ∈ 𝐶 ⟨𝑦, 𝑧⟩ ∈ (𝐴 ∘ 𝐵))
21	17, 18	opelco 4908	. . . . 5 ⊢ (⟨𝑦, 𝑧⟩ ∈ (𝐴 ∘ 𝐵) ↔ ∃𝑤(𝑦𝐵𝑤 ∧ 𝑤𝐴𝑧))
22	21	rexbii 2540	. . . 4 ⊢ (∃𝑥 ∈ 𝐶 ⟨𝑦, 𝑧⟩ ∈ (𝐴 ∘ 𝐵) ↔ ∃𝑥 ∈ 𝐶 ∃𝑤(𝑦𝐵𝑤 ∧ 𝑤𝐴𝑧))
23	20, 22	bitri 184	. . 3 ⊢ (⟨𝑦, 𝑧⟩ ∈ ∪ 𝑥 ∈ 𝐶 (𝐴 ∘ 𝐵) ↔ ∃𝑥 ∈ 𝐶 ∃𝑤(𝑦𝐵𝑤 ∧ 𝑤𝐴𝑧))
24	16, 19, 23	3bitr4i 212	. 2 ⊢ (⟨𝑦, 𝑧⟩ ∈ (𝐴 ∘ ∪ 𝑥 ∈ 𝐶 𝐵) ↔ ⟨𝑦, 𝑧⟩ ∈ ∪ 𝑥 ∈ 𝐶 (𝐴 ∘ 𝐵))
25	1, 5, 24	eqrelriiv 4826	1 ⊢ (𝐴 ∘ ∪ 𝑥 ∈ 𝐶 𝐵) = ∪ 𝑥 ∈ 𝐶 (𝐴 ∘ 𝐵)

Syntax hints:   ∧ wa 104   = wceq 1398  ∃wex 1541   ∈ wcel 2202  ∃wrex 2512  ⟨cop 3676  ∪ ciun 3975   class class class wbr 4093   ∘ ccom 4735  Rel wrel 4736

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-iun 3977  df-br 4094  df-opab 4156  df-xp 4737  df-rel 4738  df-co 4740

This theorem is referenced by: (None)