Theorem coiun 6277

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 6122	. 2 ⊢ Rel (𝐴 ∘ ∪ 𝑥 ∈ 𝐶 𝐵)
2		reliun 5803	. . 3 ⊢ (Rel ∪ 𝑥 ∈ 𝐶 (𝐴 ∘ 𝐵) ↔ ∀𝑥 ∈ 𝐶 Rel (𝐴 ∘ 𝐵))
3		relco 6122	. . . 4 ⊢ Rel (𝐴 ∘ 𝐵)
4	3	a1i 11	. . 3 ⊢ (𝑥 ∈ 𝐶 → Rel (𝐴 ∘ 𝐵))
5	2, 4	mprgbir 3087	. 2 ⊢ Rel ∪ 𝑥 ∈ 𝐶 (𝐴 ∘ 𝐵)
6		eliun 4960	. . . . . . . . 9 ⊢ (⟨𝑦, 𝑤⟩ ∈ ∪ 𝑥 ∈ 𝐶 𝐵 ↔ ∃𝑥 ∈ 𝐶 ⟨𝑦, 𝑤⟩ ∈ 𝐵)
7		df-br 5109	. . . . . . . . 9 ⊢ (𝑦∪ 𝑥 ∈ 𝐶 𝐵𝑤 ↔ ⟨𝑦, 𝑤⟩ ∈ ∪ 𝑥 ∈ 𝐶 𝐵)
8		df-br 5109	. . . . . . . . . 10 ⊢ (𝑦𝐵𝑤 ↔ ⟨𝑦, 𝑤⟩ ∈ 𝐵)
9	8	rexbii 3113	. . . . . . . . 9 ⊢ (∃𝑥 ∈ 𝐶 𝑦𝐵𝑤 ↔ ∃𝑥 ∈ 𝐶 ⟨𝑦, 𝑤⟩ ∈ 𝐵)
10	6, 7, 9	3bitr4i 306	. . . . . . . 8 ⊢ (𝑦∪ 𝑥 ∈ 𝐶 𝐵𝑤 ↔ ∃𝑥 ∈ 𝐶 𝑦𝐵𝑤)
11	10	anbi1i 636	. . . . . . 7 ⊢ ((𝑦∪ 𝑥 ∈ 𝐶 𝐵𝑤 ∧ 𝑤𝐴𝑧) ↔ (∃𝑥 ∈ 𝐶 𝑦𝐵𝑤 ∧ 𝑤𝐴𝑧))
12		r19.41v 3196	. . . . . . 7 ⊢ (∃𝑥 ∈ 𝐶 (𝑦𝐵𝑤 ∧ 𝑤𝐴𝑧) ↔ (∃𝑥 ∈ 𝐶 𝑦𝐵𝑤 ∧ 𝑤𝐴𝑧))
13	11, 12	bitr4i 281	. . . . . 6 ⊢ ((𝑦∪ 𝑥 ∈ 𝐶 𝐵𝑤 ∧ 𝑤𝐴𝑧) ↔ ∃𝑥 ∈ 𝐶 (𝑦𝐵𝑤 ∧ 𝑤𝐴𝑧))
14	13	exbii 1881	. . . . 5 ⊢ (∃𝑤(𝑦∪ 𝑥 ∈ 𝐶 𝐵𝑤 ∧ 𝑤𝐴𝑧) ↔ ∃𝑤∃𝑥 ∈ 𝐶 (𝑦𝐵𝑤 ∧ 𝑤𝐴𝑧))
15		rexcom4 3293	. . . . 5 ⊢ (∃𝑥 ∈ 𝐶 ∃𝑤(𝑦𝐵𝑤 ∧ 𝑤𝐴𝑧) ↔ ∃𝑤∃𝑥 ∈ 𝐶 (𝑦𝐵𝑤 ∧ 𝑤𝐴𝑧))
16	14, 15	bitr4i 281	. . . 4 ⊢ (∃𝑤(𝑦∪ 𝑥 ∈ 𝐶 𝐵𝑤 ∧ 𝑤𝐴𝑧) ↔ ∃𝑥 ∈ 𝐶 ∃𝑤(𝑦𝐵𝑤 ∧ 𝑤𝐴𝑧))
17		vex 3458	. . . . 5 ⊢ 𝑦 ∈ V
18		vex 3458	. . . . 5 ⊢ 𝑧 ∈ V
19	17, 18	opelco 5861	. . . 4 ⊢ (⟨𝑦, 𝑧⟩ ∈ (𝐴 ∘ ∪ 𝑥 ∈ 𝐶 𝐵) ↔ ∃𝑤(𝑦∪ 𝑥 ∈ 𝐶 𝐵𝑤 ∧ 𝑤𝐴𝑧))
20	17, 18	opelco 5861	. . . . 5 ⊢ (⟨𝑦, 𝑧⟩ ∈ (𝐴 ∘ 𝐵) ↔ ∃𝑤(𝑦𝐵𝑤 ∧ 𝑤𝐴𝑧))
21	20	rexbii 3113	. . . 4 ⊢ (∃𝑥 ∈ 𝐶 ⟨𝑦, 𝑧⟩ ∈ (𝐴 ∘ 𝐵) ↔ ∃𝑥 ∈ 𝐶 ∃𝑤(𝑦𝐵𝑤 ∧ 𝑤𝐴𝑧))
22	16, 19, 21	3bitr4i 306	. . 3 ⊢ (⟨𝑦, 𝑧⟩ ∈ (𝐴 ∘ ∪ 𝑥 ∈ 𝐶 𝐵) ↔ ∃𝑥 ∈ 𝐶 ⟨𝑦, 𝑧⟩ ∈ (𝐴 ∘ 𝐵))
23		eliun 4960	. . 3 ⊢ (⟨𝑦, 𝑧⟩ ∈ ∪ 𝑥 ∈ 𝐶 (𝐴 ∘ 𝐵) ↔ ∃𝑥 ∈ 𝐶 ⟨𝑦, 𝑧⟩ ∈ (𝐴 ∘ 𝐵))
24	22, 23	bitr4i 281	. 2 ⊢ (⟨𝑦, 𝑧⟩ ∈ (𝐴 ∘ ∪ 𝑥 ∈ 𝐶 𝐵) ↔ ⟨𝑦, 𝑧⟩ ∈ ∪ 𝑥 ∈ 𝐶 (𝐴 ∘ 𝐵))
25	1, 5, 24	eqrelriiv 5775	1 ⊢ (𝐴 ∘ ∪ 𝑥 ∈ 𝐶 𝐵) = ∪ 𝑥 ∈ 𝐶 (𝐴 ∘ 𝐵)

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2147  ∃wrex 3090  ⟨cop 4840  ∪ ciun 4956   class class class wbr 5108   ∘ ccom 5662  Rel wrel 5663

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-11 2197  ax-ext 2735  ax-sep 5255  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-iun 4958  df-br 5109  df-opab 5174  df-xp 5664  df-rel 5665  df-co 5667

This theorem is used by:  fparlem3  8162  fparlem4  8163  trclrelexplem  44972  trclfvcom  44984