Theorem dfiun2g 4972

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 2147, ax-12 2185. (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 4935	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵}
2		elisset 2818	. . . . . . . . 9 ⊢ (𝐵 ∈ 𝐶 → ∃𝑧 𝑧 = 𝐵)
3		eleq2 2825	. . . . . . . . . . . 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 1919	. . . . . . . . 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 3074	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → ∀𝑥 ∈ 𝐴 (𝑤 ∈ 𝐵 → ∃𝑧(𝑤 ∈ 𝑧 ∧ 𝑧 = 𝐵)))
9		rexim 3078	. . . . . . 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 3264	. . . . . . 7 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑧(𝑤 ∈ 𝑧 ∧ 𝑧 = 𝐵) ↔ ∃𝑧∃𝑥 ∈ 𝐴 (𝑤 ∈ 𝑧 ∧ 𝑧 = 𝐵))
12		r19.42v 3169	. . . . . . . 8 ⊢ (∃𝑥 ∈ 𝐴 (𝑤 ∈ 𝑧 ∧ 𝑧 = 𝐵) ↔ (𝑤 ∈ 𝑧 ∧ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵))
13	12	exbii 1850	. . . . . . 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 251	. . . . 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 3075	. . . . . . 7 ⊢ (∃𝑥 ∈ 𝐴 (𝑤 ∈ 𝑧 ∧ 𝑧 = 𝐵) → ∃𝑥 ∈ 𝐴 𝑤 ∈ 𝐵)
18	12, 17	sylbir 235	. . . . . 6 ⊢ ((𝑤 ∈ 𝑧 ∧ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵) → ∃𝑥 ∈ 𝐴 𝑤 ∈ 𝐵)
19	18	exlimiv 1932	. . . . 5 ⊢ (∃𝑧(𝑤 ∈ 𝑧 ∧ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵) → ∃𝑥 ∈ 𝐴 𝑤 ∈ 𝐵)
20	15, 19	impbid1 225	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → (∃𝑥 ∈ 𝐴 𝑤 ∈ 𝐵 ↔ ∃𝑧(𝑤 ∈ 𝑧 ∧ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵)))
21		vex 3433	. . . . 5 ⊢ 𝑤 ∈ V
22		eleq1w 2819	. . . . . 6 ⊢ (𝑧 = 𝑤 → (𝑧 ∈ 𝐵 ↔ 𝑤 ∈ 𝐵))
23	22	rexbidv 3161	. . . . 5 ⊢ (𝑧 = 𝑤 → (∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑤 ∈ 𝐵))
24	21, 23	elab 3622	. . . 4 ⊢ (𝑤 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵} ↔ ∃𝑥 ∈ 𝐴 𝑤 ∈ 𝐵)
25		eluni 4853	. . . . 5 ⊢ (𝑤 ∈ ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ↔ ∃𝑧(𝑤 ∈ 𝑧 ∧ 𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵}))
26		vex 3433	. . . . . . . 8 ⊢ 𝑧 ∈ V
27		eqeq1 2740	. . . . . . . . 9 ⊢ (𝑦 = 𝑧 → (𝑦 = 𝐵 ↔ 𝑧 = 𝐵))
28	27	rexbidv 3161	. . . . . . . 8 ⊢ (𝑦 = 𝑧 → (∃𝑥 ∈ 𝐴 𝑦 = 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵))
29	26, 28	elab 3622	. . . . . . 7 ⊢ (𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ↔ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵)
30	29	anbi2i 624	. . . . . 6 ⊢ ((𝑤 ∈ 𝑧 ∧ 𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵}) ↔ (𝑤 ∈ 𝑧 ∧ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵))
31	30	exbii 1850	. . . . 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 2734	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵} = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵})
35	1, 34	eqtrid 2783	1 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542  ∃wex 1781   ∈ wcel 2114  {cab 2714  ∀wral 3051  ∃wrex 3061  ∪ cuni 4850  ∪ ciun 4933

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-11 2163  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3062  df-v 3431  df-uni 4851  df-iun 4935

This theorem is referenced by:  dfiun2  4974  dfiun3g  5923  abnexg  7710  iunexg  7916  uniqs  8720  ac6num  10401  iunopn  22863  pnrmopn  23308  cncmp  23357  ptcmplem3  24019  iunmbl  25520  voliun  25521  sigaclcuni  34262  sigaclcu2  34264  sigaclci  34276  measvunilem  34356  meascnbl  34363  carsgclctunlem3  34464