Theorem dfiun2g 5033

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 2138, ax-12 2172. (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 4999	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵}
2		elisset 2816	. . . . . . . . 9 ⊢ (𝐵 ∈ 𝐶 → ∃𝑧 𝑧 = 𝐵)
3		eleq2 2823	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝐵 → (𝑤 ∈ 𝑧 ↔ 𝑤 ∈ 𝐵))
4	3	pm5.32ri 577	. . . . . . . . . . 11 ⊢ ((𝑤 ∈ 𝑧 ∧ 𝑧 = 𝐵) ↔ (𝑤 ∈ 𝐵 ∧ 𝑧 = 𝐵))
5	4	simplbi2 502	. . . . . . . . . 10 ⊢ (𝑤 ∈ 𝐵 → (𝑧 = 𝐵 → (𝑤 ∈ 𝑧 ∧ 𝑧 = 𝐵)))
6	5	eximdv 1921	. . . . . . . . 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 3084	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → ∀𝑥 ∈ 𝐴 (𝑤 ∈ 𝐵 → ∃𝑧(𝑤 ∈ 𝑧 ∧ 𝑧 = 𝐵)))
9		rexim 3088	. . . . . . 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 3286	. . . . . . 7 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑧(𝑤 ∈ 𝑧 ∧ 𝑧 = 𝐵) ↔ ∃𝑧∃𝑥 ∈ 𝐴 (𝑤 ∈ 𝑧 ∧ 𝑧 = 𝐵))
12		r19.42v 3191	. . . . . . . 8 ⊢ (∃𝑥 ∈ 𝐴 (𝑤 ∈ 𝑧 ∧ 𝑧 = 𝐵) ↔ (𝑤 ∈ 𝑧 ∧ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵))
13	12	exbii 1851	. . . . . . 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 480	. . . . . . . 8 ⊢ ((𝑤 ∈ 𝑧 ∧ 𝑧 = 𝐵) → 𝑤 ∈ 𝐵)
17	16	reximi 3085	. . . . . . 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 1934	. . . . 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 3479	. . . . 5 ⊢ 𝑤 ∈ V
22		eleq1w 2817	. . . . . 6 ⊢ (𝑧 = 𝑤 → (𝑧 ∈ 𝐵 ↔ 𝑤 ∈ 𝐵))
23	22	rexbidv 3179	. . . . 5 ⊢ (𝑧 = 𝑤 → (∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑤 ∈ 𝐵))
24	21, 23	elab 3668	. . . 4 ⊢ (𝑤 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵} ↔ ∃𝑥 ∈ 𝐴 𝑤 ∈ 𝐵)
25		eluni 4911	. . . . 5 ⊢ (𝑤 ∈ ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ↔ ∃𝑧(𝑤 ∈ 𝑧 ∧ 𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵}))
26		vex 3479	. . . . . . . 8 ⊢ 𝑧 ∈ V
27		eqeq1 2737	. . . . . . . . 9 ⊢ (𝑦 = 𝑧 → (𝑦 = 𝐵 ↔ 𝑧 = 𝐵))
28	27	rexbidv 3179	. . . . . . . 8 ⊢ (𝑦 = 𝑧 → (∃𝑥 ∈ 𝐴 𝑦 = 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵))
29	26, 28	elab 3668	. . . . . . 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 1851	. . . . 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 2731	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵} = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵})
35	1, 34	eqtrid 2785	1 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵})

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1542  ∃wex 1782   ∈ wcel 2107  {cab 2710  ∀wral 3062  ∃wrex 3071  ∪ cuni 4908  ∪ ciun 4997

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-11 2155  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-v 3477  df-uni 4909  df-iun 4999

This theorem is referenced by:  dfiun2  5036  dfiun3g  5962  abnexg  7740  iunexg  7947  uniqs  8768  ac6num  10471  iunopn  22392  pnrmopn  22839  cncmp  22888  ptcmplem3  23550  iunmbl  25062  voliun  25063  sigaclcuni  33105  sigaclcu2  33107  sigaclci  33119  measvunilem  33199  meascnbl  33206  carsgclctunlem3  33308  uniqsALTV  37187