Theorem dfiun2g 5027

Description: Alternate definition of indexed union when 𝐵 is a set. Definition 15(a) of [Suppes] p. 44. (Contributed by NM, 23-Mar-2006.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) (Proof shortened by Rohan Ridenour, 11-Aug-2023.) Avoid ax-10 2130, ax-12 2164. (Revised by SN, 11-Dec-2024.)

Assertion

Ref	Expression
dfiun2g	⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵})

Distinct variable groups:   𝑦,𝐴   𝑦,𝐵   𝑥,𝑦

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝐶(𝑥,𝑦)

Proof of Theorem dfiun2g

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

Step	Hyp	Ref	Expression
1		df-iun 4993	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵}
2		elisset 2810	. . . . . . . . 9 ⊢ (𝐵 ∈ 𝐶 → ∃𝑧 𝑧 = 𝐵)
3		eleq2 2817	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝐵 → (𝑤 ∈ 𝑧 ↔ 𝑤 ∈ 𝐵))
4	3	pm5.32ri 575	. . . . . . . . . . 11 ⊢ ((𝑤 ∈ 𝑧 ∧ 𝑧 = 𝐵) ↔ (𝑤 ∈ 𝐵 ∧ 𝑧 = 𝐵))
5	4	simplbi2 500	. . . . . . . . . 10 ⊢ (𝑤 ∈ 𝐵 → (𝑧 = 𝐵 → (𝑤 ∈ 𝑧 ∧ 𝑧 = 𝐵)))
6	5	eximdv 1913	. . . . . . . . 9 ⊢ (𝑤 ∈ 𝐵 → (∃𝑧 𝑧 = 𝐵 → ∃𝑧(𝑤 ∈ 𝑧 ∧ 𝑧 = 𝐵)))
7	2, 6	syl5com 31	. . . . . . . 8 ⊢ (𝐵 ∈ 𝐶 → (𝑤 ∈ 𝐵 → ∃𝑧(𝑤 ∈ 𝑧 ∧ 𝑧 = 𝐵)))
8	7	ralimi 3078	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → ∀𝑥 ∈ 𝐴 (𝑤 ∈ 𝐵 → ∃𝑧(𝑤 ∈ 𝑧 ∧ 𝑧 = 𝐵)))
9		rexim 3082	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 (𝑤 ∈ 𝐵 → ∃𝑧(𝑤 ∈ 𝑧 ∧ 𝑧 = 𝐵)) → (∃𝑥 ∈ 𝐴 𝑤 ∈ 𝐵 → ∃𝑥 ∈ 𝐴 ∃𝑧(𝑤 ∈ 𝑧 ∧ 𝑧 = 𝐵)))
10	8, 9	syl 17	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → (∃𝑥 ∈ 𝐴 𝑤 ∈ 𝐵 → ∃𝑥 ∈ 𝐴 ∃𝑧(𝑤 ∈ 𝑧 ∧ 𝑧 = 𝐵)))
11		rexcom4 3280	. . . . . . 7 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑧(𝑤 ∈ 𝑧 ∧ 𝑧 = 𝐵) ↔ ∃𝑧∃𝑥 ∈ 𝐴 (𝑤 ∈ 𝑧 ∧ 𝑧 = 𝐵))
12		r19.42v 3185	. . . . . . . 8 ⊢ (∃𝑥 ∈ 𝐴 (𝑤 ∈ 𝑧 ∧ 𝑧 = 𝐵) ↔ (𝑤 ∈ 𝑧 ∧ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵))
13	12	exbii 1843	. . . . . . 7 ⊢ (∃𝑧∃𝑥 ∈ 𝐴 (𝑤 ∈ 𝑧 ∧ 𝑧 = 𝐵) ↔ ∃𝑧(𝑤 ∈ 𝑧 ∧ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵))
14	11, 13	bitri 275	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑧(𝑤 ∈ 𝑧 ∧ 𝑧 = 𝐵) ↔ ∃𝑧(𝑤 ∈ 𝑧 ∧ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵))
15	10, 14	imbitrdi 250	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → (∃𝑥 ∈ 𝐴 𝑤 ∈ 𝐵 → ∃𝑧(𝑤 ∈ 𝑧 ∧ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵)))
16	3	biimpac 478	. . . . . . . 8 ⊢ ((𝑤 ∈ 𝑧 ∧ 𝑧 = 𝐵) → 𝑤 ∈ 𝐵)
17	16	reximi 3079	. . . . . . 7 ⊢ (∃𝑥 ∈ 𝐴 (𝑤 ∈ 𝑧 ∧ 𝑧 = 𝐵) → ∃𝑥 ∈ 𝐴 𝑤 ∈ 𝐵)
18	12, 17	sylbir 234	. . . . . 6 ⊢ ((𝑤 ∈ 𝑧 ∧ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵) → ∃𝑥 ∈ 𝐴 𝑤 ∈ 𝐵)
19	18	exlimiv 1926	. . . . 5 ⊢ (∃𝑧(𝑤 ∈ 𝑧 ∧ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵) → ∃𝑥 ∈ 𝐴 𝑤 ∈ 𝐵)
20	15, 19	impbid1 224	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → (∃𝑥 ∈ 𝐴 𝑤 ∈ 𝐵 ↔ ∃𝑧(𝑤 ∈ 𝑧 ∧ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵)))
21		vex 3473	. . . . 5 ⊢ 𝑤 ∈ V
22		eleq1w 2811	. . . . . 6 ⊢ (𝑧 = 𝑤 → (𝑧 ∈ 𝐵 ↔ 𝑤 ∈ 𝐵))
23	22	rexbidv 3173	. . . . 5 ⊢ (𝑧 = 𝑤 → (∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑤 ∈ 𝐵))
24	21, 23	elab 3665	. . . 4 ⊢ (𝑤 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵} ↔ ∃𝑥 ∈ 𝐴 𝑤 ∈ 𝐵)
25		eluni 4906	. . . . 5 ⊢ (𝑤 ∈ ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ↔ ∃𝑧(𝑤 ∈ 𝑧 ∧ 𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵}))
26		vex 3473	. . . . . . . 8 ⊢ 𝑧 ∈ V
27		eqeq1 2731	. . . . . . . . 9 ⊢ (𝑦 = 𝑧 → (𝑦 = 𝐵 ↔ 𝑧 = 𝐵))
28	27	rexbidv 3173	. . . . . . . 8 ⊢ (𝑦 = 𝑧 → (∃𝑥 ∈ 𝐴 𝑦 = 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵))
29	26, 28	elab 3665	. . . . . . 7 ⊢ (𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ↔ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵)
30	29	anbi2i 622	. . . . . 6 ⊢ ((𝑤 ∈ 𝑧 ∧ 𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵}) ↔ (𝑤 ∈ 𝑧 ∧ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵))
31	30	exbii 1843	. . . . 5 ⊢ (∃𝑧(𝑤 ∈ 𝑧 ∧ 𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵}) ↔ ∃𝑧(𝑤 ∈ 𝑧 ∧ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵))
32	25, 31	bitri 275	. . . 4 ⊢ (𝑤 ∈ ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ↔ ∃𝑧(𝑤 ∈ 𝑧 ∧ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵))
33	20, 24, 32	3bitr4g 314	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → (𝑤 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵} ↔ 𝑤 ∈ ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵}))
34	33	eqrdv 2725	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵} = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵})
35	1, 34	eqtrid 2779	1 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1534  ∃wex 1774   ∈ wcel 2099  {cab 2704  ∀wral 3056  ∃wrex 3065  ∪ cuni 4903  ∪ ciun 4991

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-11 2147  ax-ext 2698

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-ral 3057  df-rex 3066  df-v 3471  df-uni 4904  df-iun 4993

This theorem is referenced by:  dfiun2  5030  dfiun3g  5961  abnexg  7752  iunexg  7961  uniqs  8787  ac6num  10494  iunopn  22787  pnrmopn  23234  cncmp  23283  ptcmplem3  23945  iunmbl  25469  voliun  25470  sigaclcuni  33673  sigaclcu2  33675  sigaclci  33687  measvunilem  33767  meascnbl  33774  carsgclctunlem3  33876  uniqsALTV  37737