Theorem coiun 5048

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 5037	. 2 ⊢ Rel (𝐴 ∘ ∪ 𝑥 ∈ 𝐶 𝐵)
2		reliun 4660	. . 3 ⊢ (Rel ∪ 𝑥 ∈ 𝐶 (𝐴 ∘ 𝐵) ↔ ∀𝑥 ∈ 𝐶 Rel (𝐴 ∘ 𝐵))
3		relco 5037	. . . 4 ⊢ Rel (𝐴 ∘ 𝐵)
4	3	a1i 9	. . 3 ⊢ (𝑥 ∈ 𝐶 → Rel (𝐴 ∘ 𝐵))
5	2, 4	mprgbir 2490	. 2 ⊢ Rel ∪ 𝑥 ∈ 𝐶 (𝐴 ∘ 𝐵)
6		eliun 3817	. . . . . . . 8 ⊢ (⟨𝑦, 𝑤⟩ ∈ ∪ 𝑥 ∈ 𝐶 𝐵 ↔ ∃𝑥 ∈ 𝐶 ⟨𝑦, 𝑤⟩ ∈ 𝐵)
7		df-br 3930	. . . . . . . 8 ⊢ (𝑦∪ 𝑥 ∈ 𝐶 𝐵𝑤 ↔ ⟨𝑦, 𝑤⟩ ∈ ∪ 𝑥 ∈ 𝐶 𝐵)
8		df-br 3930	. . . . . . . . 9 ⊢ (𝑦𝐵𝑤 ↔ ⟨𝑦, 𝑤⟩ ∈ 𝐵)
9	8	rexbii 2442	. . . . . . . 8 ⊢ (∃𝑥 ∈ 𝐶 𝑦𝐵𝑤 ↔ ∃𝑥 ∈ 𝐶 ⟨𝑦, 𝑤⟩ ∈ 𝐵)
10	6, 7, 9	3bitr4i 211	. . . . . . 7 ⊢ (𝑦∪ 𝑥 ∈ 𝐶 𝐵𝑤 ↔ ∃𝑥 ∈ 𝐶 𝑦𝐵𝑤)
11	10	anbi1i 453	. . . . . 6 ⊢ ((𝑦∪ 𝑥 ∈ 𝐶 𝐵𝑤 ∧ 𝑤𝐴𝑧) ↔ (∃𝑥 ∈ 𝐶 𝑦𝐵𝑤 ∧ 𝑤𝐴𝑧))
12		r19.41v 2587	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐶 (𝑦𝐵𝑤 ∧ 𝑤𝐴𝑧) ↔ (∃𝑥 ∈ 𝐶 𝑦𝐵𝑤 ∧ 𝑤𝐴𝑧))
13	11, 12	bitr4i 186	. . . . 5 ⊢ ((𝑦∪ 𝑥 ∈ 𝐶 𝐵𝑤 ∧ 𝑤𝐴𝑧) ↔ ∃𝑥 ∈ 𝐶 (𝑦𝐵𝑤 ∧ 𝑤𝐴𝑧))
14	13	exbii 1584	. . . 4 ⊢ (∃𝑤(𝑦∪ 𝑥 ∈ 𝐶 𝐵𝑤 ∧ 𝑤𝐴𝑧) ↔ ∃𝑤∃𝑥 ∈ 𝐶 (𝑦𝐵𝑤 ∧ 𝑤𝐴𝑧))
15		rexcom4 2709	. . . 4 ⊢ (∃𝑥 ∈ 𝐶 ∃𝑤(𝑦𝐵𝑤 ∧ 𝑤𝐴𝑧) ↔ ∃𝑤∃𝑥 ∈ 𝐶 (𝑦𝐵𝑤 ∧ 𝑤𝐴𝑧))
16	14, 15	bitr4i 186	. . 3 ⊢ (∃𝑤(𝑦∪ 𝑥 ∈ 𝐶 𝐵𝑤 ∧ 𝑤𝐴𝑧) ↔ ∃𝑥 ∈ 𝐶 ∃𝑤(𝑦𝐵𝑤 ∧ 𝑤𝐴𝑧))
17		vex 2689	. . . 4 ⊢ 𝑦 ∈ V
18		vex 2689	. . . 4 ⊢ 𝑧 ∈ V
19	17, 18	opelco 4711	. . 3 ⊢ (⟨𝑦, 𝑧⟩ ∈ (𝐴 ∘ ∪ 𝑥 ∈ 𝐶 𝐵) ↔ ∃𝑤(𝑦∪ 𝑥 ∈ 𝐶 𝐵𝑤 ∧ 𝑤𝐴𝑧))
20		eliun 3817	. . . 4 ⊢ (⟨𝑦, 𝑧⟩ ∈ ∪ 𝑥 ∈ 𝐶 (𝐴 ∘ 𝐵) ↔ ∃𝑥 ∈ 𝐶 ⟨𝑦, 𝑧⟩ ∈ (𝐴 ∘ 𝐵))
21	17, 18	opelco 4711	. . . . 5 ⊢ (⟨𝑦, 𝑧⟩ ∈ (𝐴 ∘ 𝐵) ↔ ∃𝑤(𝑦𝐵𝑤 ∧ 𝑤𝐴𝑧))
22	21	rexbii 2442	. . . 4 ⊢ (∃𝑥 ∈ 𝐶 ⟨𝑦, 𝑧⟩ ∈ (𝐴 ∘ 𝐵) ↔ ∃𝑥 ∈ 𝐶 ∃𝑤(𝑦𝐵𝑤 ∧ 𝑤𝐴𝑧))
23	20, 22	bitri 183	. . 3 ⊢ (⟨𝑦, 𝑧⟩ ∈ ∪ 𝑥 ∈ 𝐶 (𝐴 ∘ 𝐵) ↔ ∃𝑥 ∈ 𝐶 ∃𝑤(𝑦𝐵𝑤 ∧ 𝑤𝐴𝑧))
24	16, 19, 23	3bitr4i 211	. 2 ⊢ (⟨𝑦, 𝑧⟩ ∈ (𝐴 ∘ ∪ 𝑥 ∈ 𝐶 𝐵) ↔ ⟨𝑦, 𝑧⟩ ∈ ∪ 𝑥 ∈ 𝐶 (𝐴 ∘ 𝐵))
25	1, 5, 24	eqrelriiv 4633	1 ⊢ (𝐴 ∘ ∪ 𝑥 ∈ 𝐶 𝐵) = ∪ 𝑥 ∈ 𝐶 (𝐴 ∘ 𝐵)

Syntax hints:   ∧ wa 103   = wceq 1331  ∃wex 1468   ∈ wcel 1480  ∃wrex 2417  ⟨cop 3530  ∪ ciun 3813   class class class wbr 3929   ∘ ccom 4543  Rel wrel 4544

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-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-iun 3815  df-br 3930  df-opab 3990  df-xp 4545  df-rel 4546  df-co 4548

This theorem is referenced by: (None)