Theorem coiun1 43648

Description: Composition with an indexed union. Proof analogous to that of coiun 6232. (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 6082	. 2 ⊢ Rel (∪ 𝑥 ∈ 𝐶 𝐴 ∘ 𝐵)
2		reliun 5782	. . 3 ⊢ (Rel ∪ 𝑥 ∈ 𝐶 (𝐴 ∘ 𝐵) ↔ ∀𝑥 ∈ 𝐶 Rel (𝐴 ∘ 𝐵))
3		relco 6082	. . . 4 ⊢ Rel (𝐴 ∘ 𝐵)
4	3	a1i 11	. . 3 ⊢ (𝑥 ∈ 𝐶 → Rel (𝐴 ∘ 𝐵))
5	2, 4	mprgbir 3052	. 2 ⊢ Rel ∪ 𝑥 ∈ 𝐶 (𝐴 ∘ 𝐵)
6		eliun 4962	. . . . . . . 8 ⊢ (⟨𝑤, 𝑧⟩ ∈ ∪ 𝑥 ∈ 𝐶 𝐴 ↔ ∃𝑥 ∈ 𝐶 ⟨𝑤, 𝑧⟩ ∈ 𝐴)
7		df-br 5111	. . . . . . . 8 ⊢ (𝑤∪ 𝑥 ∈ 𝐶 𝐴𝑧 ↔ ⟨𝑤, 𝑧⟩ ∈ ∪ 𝑥 ∈ 𝐶 𝐴)
8		df-br 5111	. . . . . . . . 9 ⊢ (𝑤𝐴𝑧 ↔ ⟨𝑤, 𝑧⟩ ∈ 𝐴)
9	8	rexbii 3077	. . . . . . . 8 ⊢ (∃𝑥 ∈ 𝐶 𝑤𝐴𝑧 ↔ ∃𝑥 ∈ 𝐶 ⟨𝑤, 𝑧⟩ ∈ 𝐴)
10	6, 7, 9	3bitr4i 303	. . . . . . 7 ⊢ (𝑤∪ 𝑥 ∈ 𝐶 𝐴𝑧 ↔ ∃𝑥 ∈ 𝐶 𝑤𝐴𝑧)
11	10	anbi2i 623	. . . . . 6 ⊢ ((𝑦𝐵𝑤 ∧ 𝑤∪ 𝑥 ∈ 𝐶 𝐴𝑧) ↔ (𝑦𝐵𝑤 ∧ ∃𝑥 ∈ 𝐶 𝑤𝐴𝑧))
12		r19.42v 3170	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐶 (𝑦𝐵𝑤 ∧ 𝑤𝐴𝑧) ↔ (𝑦𝐵𝑤 ∧ ∃𝑥 ∈ 𝐶 𝑤𝐴𝑧))
13	11, 12	bitr4i 278	. . . . 5 ⊢ ((𝑦𝐵𝑤 ∧ 𝑤∪ 𝑥 ∈ 𝐶 𝐴𝑧) ↔ ∃𝑥 ∈ 𝐶 (𝑦𝐵𝑤 ∧ 𝑤𝐴𝑧))
14	13	exbii 1848	. . . 4 ⊢ (∃𝑤(𝑦𝐵𝑤 ∧ 𝑤∪ 𝑥 ∈ 𝐶 𝐴𝑧) ↔ ∃𝑤∃𝑥 ∈ 𝐶 (𝑦𝐵𝑤 ∧ 𝑤𝐴𝑧))
15		rexcom4 3265	. . . 4 ⊢ (∃𝑥 ∈ 𝐶 ∃𝑤(𝑦𝐵𝑤 ∧ 𝑤𝐴𝑧) ↔ ∃𝑤∃𝑥 ∈ 𝐶 (𝑦𝐵𝑤 ∧ 𝑤𝐴𝑧))
16	14, 15	bitr4i 278	. . 3 ⊢ (∃𝑤(𝑦𝐵𝑤 ∧ 𝑤∪ 𝑥 ∈ 𝐶 𝐴𝑧) ↔ ∃𝑥 ∈ 𝐶 ∃𝑤(𝑦𝐵𝑤 ∧ 𝑤𝐴𝑧))
17		vex 3454	. . . 4 ⊢ 𝑦 ∈ V
18		vex 3454	. . . 4 ⊢ 𝑧 ∈ V
19	17, 18	opelco 5838	. . 3 ⊢ (⟨𝑦, 𝑧⟩ ∈ (∪ 𝑥 ∈ 𝐶 𝐴 ∘ 𝐵) ↔ ∃𝑤(𝑦𝐵𝑤 ∧ 𝑤∪ 𝑥 ∈ 𝐶 𝐴𝑧))
20		eliun 4962	. . . 4 ⊢ (⟨𝑦, 𝑧⟩ ∈ ∪ 𝑥 ∈ 𝐶 (𝐴 ∘ 𝐵) ↔ ∃𝑥 ∈ 𝐶 ⟨𝑦, 𝑧⟩ ∈ (𝐴 ∘ 𝐵))
21	17, 18	opelco 5838	. . . . 5 ⊢ (⟨𝑦, 𝑧⟩ ∈ (𝐴 ∘ 𝐵) ↔ ∃𝑤(𝑦𝐵𝑤 ∧ 𝑤𝐴𝑧))
22	21	rexbii 3077	. . . 4 ⊢ (∃𝑥 ∈ 𝐶 ⟨𝑦, 𝑧⟩ ∈ (𝐴 ∘ 𝐵) ↔ ∃𝑥 ∈ 𝐶 ∃𝑤(𝑦𝐵𝑤 ∧ 𝑤𝐴𝑧))
23	20, 22	bitri 275	. . 3 ⊢ (⟨𝑦, 𝑧⟩ ∈ ∪ 𝑥 ∈ 𝐶 (𝐴 ∘ 𝐵) ↔ ∃𝑥 ∈ 𝐶 ∃𝑤(𝑦𝐵𝑤 ∧ 𝑤𝐴𝑧))
24	16, 19, 23	3bitr4i 303	. 2 ⊢ (⟨𝑦, 𝑧⟩ ∈ (∪ 𝑥 ∈ 𝐶 𝐴 ∘ 𝐵) ↔ ⟨𝑦, 𝑧⟩ ∈ ∪ 𝑥 ∈ 𝐶 (𝐴 ∘ 𝐵))
25	1, 5, 24	eqrelriiv 5756	1 ⊢ (∪ 𝑥 ∈ 𝐶 𝐴 ∘ 𝐵) = ∪ 𝑥 ∈ 𝐶 (𝐴 ∘ 𝐵)

Syntax hints:   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109  ∃wrex 3054  ⟨cop 4598  ∪ ciun 4958   class class class wbr 5110   ∘ ccom 5645  Rel wrel 5646

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-iun 4960  df-br 5111  df-opab 5173  df-xp 5647  df-rel 5648  df-co 5650

This theorem is referenced by:  trclfvcom  43719  trclfvdecomr  43724  cotrclrcl  43738