Theorem coiun1 44330

Description: Composition with an indexed union. Proof analogous to that of coiun 6262. (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 6114	. 2 ⊢ Rel (∪ 𝑥 ∈ 𝐶 𝐴 ∘ 𝐵)
2		reliun 5807	. . 3 ⊢ (Rel ∪ 𝑥 ∈ 𝐶 (𝐴 ∘ 𝐵) ↔ ∀𝑥 ∈ 𝐶 Rel (𝐴 ∘ 𝐵))
3		relco 6114	. . . 4 ⊢ Rel (𝐴 ∘ 𝐵)
4	3	a1i 11	. . 3 ⊢ (𝑥 ∈ 𝐶 → Rel (𝐴 ∘ 𝐵))
5	2, 4	mprgbir 3093	. 2 ⊢ Rel ∪ 𝑥 ∈ 𝐶 (𝐴 ∘ 𝐵)
6		eliun 4965	. . . . . . . 8 ⊢ (⟨𝑤, 𝑧⟩ ∈ ∪ 𝑥 ∈ 𝐶 𝐴 ↔ ∃𝑥 ∈ 𝐶 ⟨𝑤, 𝑧⟩ ∈ 𝐴)
7		df-br 5115	. . . . . . . 8 ⊢ (𝑤∪ 𝑥 ∈ 𝐶 𝐴𝑧 ↔ ⟨𝑤, 𝑧⟩ ∈ ∪ 𝑥 ∈ 𝐶 𝐴)
8		df-br 5115	. . . . . . . . 9 ⊢ (𝑤𝐴𝑧 ↔ ⟨𝑤, 𝑧⟩ ∈ 𝐴)
9	8	rexbii 3119	. . . . . . . 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 634	. . . . . 6 ⊢ ((𝑦𝐵𝑤 ∧ 𝑤∪ 𝑥 ∈ 𝐶 𝐴𝑧) ↔ (𝑦𝐵𝑤 ∧ ∃𝑥 ∈ 𝐶 𝑤𝐴𝑧))
12		r19.42v 3204	. . . . . 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 1876	. . . 4 ⊢ (∃𝑤(𝑦𝐵𝑤 ∧ 𝑤∪ 𝑥 ∈ 𝐶 𝐴𝑧) ↔ ∃𝑤∃𝑥 ∈ 𝐶 (𝑦𝐵𝑤 ∧ 𝑤𝐴𝑧))
15		rexcom4 3299	. . . 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 3466	. . . 4 ⊢ 𝑦 ∈ V
18		vex 3466	. . . 4 ⊢ 𝑧 ∈ V
19	17, 18	opelco 5861	. . 3 ⊢ (⟨𝑦, 𝑧⟩ ∈ (∪ 𝑥 ∈ 𝐶 𝐴 ∘ 𝐵) ↔ ∃𝑤(𝑦𝐵𝑤 ∧ 𝑤∪ 𝑥 ∈ 𝐶 𝐴𝑧))
20		eliun 4965	. . . 4 ⊢ (⟨𝑦, 𝑧⟩ ∈ ∪ 𝑥 ∈ 𝐶 (𝐴 ∘ 𝐵) ↔ ∃𝑥 ∈ 𝐶 ⟨𝑦, 𝑧⟩ ∈ (𝐴 ∘ 𝐵))
21	17, 18	opelco 5861	. . . . 5 ⊢ (⟨𝑦, 𝑧⟩ ∈ (𝐴 ∘ 𝐵) ↔ ∃𝑤(𝑦𝐵𝑤 ∧ 𝑤𝐴𝑧))
22	21	rexbii 3119	. . . 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 5780	1 ⊢ (∪ 𝑥 ∈ 𝐶 𝐴 ∘ 𝐵) = ∪ 𝑥 ∈ 𝐶 (𝐴 ∘ 𝐵)

Syntax hints:   ∧ wa 400   = wceq 1568  ∃wex 1807   ∈ wcel 2150  ∃wrex 3096  ⟨cop 4600  ∪ ciun 4961   class class class wbr 5114   ∘ ccom 5669  Rel wrel 5670

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-11 2199  ax-ext 2742  ax-sep 5262  ax-pr 5408

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-sb 2099  df-clab 2749  df-cleq 2762  df-clel 2845  df-ral 3087  df-rex 3097  df-rab 3424  df-v 3464  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-iun 4963  df-br 5115  df-opab 5179  df-xp 5671  df-rel 5672  df-co 5674

This theorem is referenced by:  trclfvcom  44401  trclfvdecomr  44406  cotrclrcl  44420