Theorem coiun1 44479

Description: Composition with an indexed union. Proof analogous to that of coiun 6257. (Contributed by RP, 20-Jun-2020.)

Assertion

Ref	Expression
coiun1	⊢ (∪ 𝑥 ∈ 𝐶 𝐴 ∘ 𝐵) = ∪ 𝑥 ∈ 𝐶 (𝐴 ∘ 𝐵)

Distinct variable group:   𝑥,𝐵

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

Proof of Theorem coiun1

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

Step	Hyp	Ref	Expression
1		relco 6108	. 2 ⊢ Rel (∪ 𝑥 ∈ 𝐶 𝐴 ∘ 𝐵)
2		reliun 5801	. . 3 ⊢ (Rel ∪ 𝑥 ∈ 𝐶 (𝐴 ∘ 𝐵) ↔ ∀𝑥 ∈ 𝐶 Rel (𝐴 ∘ 𝐵))
3		relco 6108	. . . 4 ⊢ Rel (𝐴 ∘ 𝐵)
4	3	a1i 11	. . 3 ⊢ (𝑥 ∈ 𝐶 → Rel (𝐴 ∘ 𝐵))
5	2, 4	mprgbir 3085	. 2 ⊢ Rel ∪ 𝑥 ∈ 𝐶 (𝐴 ∘ 𝐵)
6		eliun 4958	. . . . . . . 8 ⊢ (⟨𝑤, 𝑧⟩ ∈ ∪ 𝑥 ∈ 𝐶 𝐴 ↔ ∃𝑥 ∈ 𝐶 ⟨𝑤, 𝑧⟩ ∈ 𝐴)
7		df-br 5108	. . . . . . . 8 ⊢ (𝑤∪ 𝑥 ∈ 𝐶 𝐴𝑧 ↔ ⟨𝑤, 𝑧⟩ ∈ ∪ 𝑥 ∈ 𝐶 𝐴)
8		df-br 5108	. . . . . . . . 9 ⊢ (𝑤𝐴𝑧 ↔ ⟨𝑤, 𝑧⟩ ∈ 𝐴)
9	8	rexbii 3111	. . . . . . . 8 ⊢ (∃𝑥 ∈ 𝐶 𝑤𝐴𝑧 ↔ ∃𝑥 ∈ 𝐶 ⟨𝑤, 𝑧⟩ ∈ 𝐴)
10	6, 7, 9	3bitr4i 306	. . . . . . 7 ⊢ (𝑤∪ 𝑥 ∈ 𝐶 𝐴𝑧 ↔ ∃𝑥 ∈ 𝐶 𝑤𝐴𝑧)
11	10	anbi2i 635	. . . . . 6 ⊢ ((𝑦𝐵𝑤 ∧ 𝑤∪ 𝑥 ∈ 𝐶 𝐴𝑧) ↔ (𝑦𝐵𝑤 ∧ ∃𝑥 ∈ 𝐶 𝑤𝐴𝑧))
12		r19.42v 3196	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐶 (𝑦𝐵𝑤 ∧ 𝑤𝐴𝑧) ↔ (𝑦𝐵𝑤 ∧ ∃𝑥 ∈ 𝐶 𝑤𝐴𝑧))
13	11, 12	bitr4i 281	. . . . 5 ⊢ ((𝑦𝐵𝑤 ∧ 𝑤∪ 𝑥 ∈ 𝐶 𝐴𝑧) ↔ ∃𝑥 ∈ 𝐶 (𝑦𝐵𝑤 ∧ 𝑤𝐴𝑧))
14	13	exbii 1881	. . . 4 ⊢ (∃𝑤(𝑦𝐵𝑤 ∧ 𝑤∪ 𝑥 ∈ 𝐶 𝐴𝑧) ↔ ∃𝑤∃𝑥 ∈ 𝐶 (𝑦𝐵𝑤 ∧ 𝑤𝐴𝑧))
15		rexcom4 3291	. . . 4 ⊢ (∃𝑥 ∈ 𝐶 ∃𝑤(𝑦𝐵𝑤 ∧ 𝑤𝐴𝑧) ↔ ∃𝑤∃𝑥 ∈ 𝐶 (𝑦𝐵𝑤 ∧ 𝑤𝐴𝑧))
16	14, 15	bitr4i 281	. . 3 ⊢ (∃𝑤(𝑦𝐵𝑤 ∧ 𝑤∪ 𝑥 ∈ 𝐶 𝐴𝑧) ↔ ∃𝑥 ∈ 𝐶 ∃𝑤(𝑦𝐵𝑤 ∧ 𝑤𝐴𝑧))
17		vex 3457	. . . 4 ⊢ 𝑦 ∈ V
18		vex 3457	. . . 4 ⊢ 𝑧 ∈ V
19	17, 18	opelco 5855	. . 3 ⊢ (⟨𝑦, 𝑧⟩ ∈ (∪ 𝑥 ∈ 𝐶 𝐴 ∘ 𝐵) ↔ ∃𝑤(𝑦𝐵𝑤 ∧ 𝑤∪ 𝑥 ∈ 𝐶 𝐴𝑧))
20		eliun 4958	. . . 4 ⊢ (⟨𝑦, 𝑧⟩ ∈ ∪ 𝑥 ∈ 𝐶 (𝐴 ∘ 𝐵) ↔ ∃𝑥 ∈ 𝐶 ⟨𝑦, 𝑧⟩ ∈ (𝐴 ∘ 𝐵))
21	17, 18	opelco 5855	. . . . 5 ⊢ (⟨𝑦, 𝑧⟩ ∈ (𝐴 ∘ 𝐵) ↔ ∃𝑤(𝑦𝐵𝑤 ∧ 𝑤𝐴𝑧))
22	21	rexbii 3111	. . . 4 ⊢ (∃𝑥 ∈ 𝐶 ⟨𝑦, 𝑧⟩ ∈ (𝐴 ∘ 𝐵) ↔ ∃𝑥 ∈ 𝐶 ∃𝑤(𝑦𝐵𝑤 ∧ 𝑤𝐴𝑧))
23	20, 22	bitri 278	. . 3 ⊢ (⟨𝑦, 𝑧⟩ ∈ ∪ 𝑥 ∈ 𝐶 (𝐴 ∘ 𝐵) ↔ ∃𝑥 ∈ 𝐶 ∃𝑤(𝑦𝐵𝑤 ∧ 𝑤𝐴𝑧))
24	16, 19, 23	3bitr4i 306	. 2 ⊢ (⟨𝑦, 𝑧⟩ ∈ (∪ 𝑥 ∈ 𝐶 𝐴 ∘ 𝐵) ↔ ⟨𝑦, 𝑧⟩ ∈ ∪ 𝑥 ∈ 𝐶 (𝐴 ∘ 𝐵))
25	1, 5, 24	eqrelriiv 5774	1 ⊢ (∪ 𝑥 ∈ 𝐶 𝐴 ∘ 𝐵) = ∪ 𝑥 ∈ 𝐶 (𝐴 ∘ 𝐵)

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2145  ∃wrex 3088  ⟨cop 4593  ∪ ciun 4954   class class class wbr 5107   ∘ ccom 5663  Rel wrel 5664

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-11 2194  ax-ext 2734  ax-sep 5255  ax-pr 5402

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 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-iun 4956  df-br 5108  df-opab 5172  df-xp 5665  df-rel 5666  df-co 5668

This theorem is used by:  trclfvcom  44550  trclfvdecomr  44555  cotrclrcl  44569