Theorem dfiun2g 4998

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 2182, ax-12 2219. (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 4962	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵}
2		elisset 2851	. . . . . . . . 9 ⊢ (𝐵 ∈ 𝐶 → ∃𝑧 𝑧 = 𝐵)
3		eleq2 2858	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝐵 → (𝑤 ∈ 𝑧 ↔ 𝑤 ∈ 𝐵))
4	3	pm5.32ri 585	. . . . . . . . . . 11 ⊢ ((𝑤 ∈ 𝑧 ∧ 𝑧 = 𝐵) ↔ (𝑤 ∈ 𝐵 ∧ 𝑧 = 𝐵))
5	4	simplbi2 505	. . . . . . . . . 10 ⊢ (𝑤 ∈ 𝐵 → (𝑧 = 𝐵 → (𝑤 ∈ 𝑧 ∧ 𝑧 = 𝐵)))
6	5	eximdv 1944	. . . . . . . . 9 ⊢ (𝑤 ∈ 𝐵 → (∃𝑧 𝑧 = 𝐵 → ∃𝑧(𝑤 ∈ 𝑧 ∧ 𝑧 = 𝐵)))
7	2, 6	syl5com 32	. . . . . . . 8 ⊢ (𝐵 ∈ 𝐶 → (𝑤 ∈ 𝐵 → ∃𝑧(𝑤 ∈ 𝑧 ∧ 𝑧 = 𝐵)))
8	7	ralimi 3108	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → ∀𝑥 ∈ 𝐴 (𝑤 ∈ 𝐵 → ∃𝑧(𝑤 ∈ 𝑧 ∧ 𝑧 = 𝐵)))
9		rexim 3112	. . . . . . 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 18	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → (∃𝑥 ∈ 𝐴 𝑤 ∈ 𝐵 → ∃𝑥 ∈ 𝐴 ∃𝑧(𝑤 ∈ 𝑧 ∧ 𝑧 = 𝐵)))
11		rexcom4 3298	. . . . . . 7 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑧(𝑤 ∈ 𝑧 ∧ 𝑧 = 𝐵) ↔ ∃𝑧∃𝑥 ∈ 𝐴 (𝑤 ∈ 𝑧 ∧ 𝑧 = 𝐵))
12		r19.42v 3203	. . . . . . . 8 ⊢ (∃𝑥 ∈ 𝐴 (𝑤 ∈ 𝑧 ∧ 𝑧 = 𝐵) ↔ (𝑤 ∈ 𝑧 ∧ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵))
13	12	exbii 1875	. . . . . . 7 ⊢ (∃𝑧∃𝑥 ∈ 𝐴 (𝑤 ∈ 𝑧 ∧ 𝑧 = 𝐵) ↔ ∃𝑧(𝑤 ∈ 𝑧 ∧ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵))
14	11, 13	bitri 278	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑧(𝑤 ∈ 𝑧 ∧ 𝑧 = 𝐵) ↔ ∃𝑧(𝑤 ∈ 𝑧 ∧ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵))
15	10, 14	imbitrdi 254	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → (∃𝑥 ∈ 𝐴 𝑤 ∈ 𝐵 → ∃𝑧(𝑤 ∈ 𝑧 ∧ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵)))
16	3	biimpac 483	. . . . . . . 8 ⊢ ((𝑤 ∈ 𝑧 ∧ 𝑧 = 𝐵) → 𝑤 ∈ 𝐵)
17	16	reximi 3109	. . . . . . 7 ⊢ (∃𝑥 ∈ 𝐴 (𝑤 ∈ 𝑧 ∧ 𝑧 = 𝐵) → ∃𝑥 ∈ 𝐴 𝑤 ∈ 𝐵)
18	12, 17	sylbir 238	. . . . . 6 ⊢ ((𝑤 ∈ 𝑧 ∧ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵) → ∃𝑥 ∈ 𝐴 𝑤 ∈ 𝐵)
19	18	exlimiv 1957	. . . . 5 ⊢ (∃𝑧(𝑤 ∈ 𝑧 ∧ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵) → ∃𝑥 ∈ 𝐴 𝑤 ∈ 𝐵)
20	15, 19	impbid1 228	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → (∃𝑥 ∈ 𝐴 𝑤 ∈ 𝐵 ↔ ∃𝑧(𝑤 ∈ 𝑧 ∧ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵)))
21		vex 3467	. . . . 5 ⊢ 𝑤 ∈ V
22		eleq1w 2852	. . . . . 6 ⊢ (𝑧 = 𝑤 → (𝑧 ∈ 𝐵 ↔ 𝑤 ∈ 𝐵))
23	22	rexbidv 3195	. . . . 5 ⊢ (𝑧 = 𝑤 → (∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑤 ∈ 𝐵))
24	21, 23	elab 3647	. . . 4 ⊢ (𝑤 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵} ↔ ∃𝑥 ∈ 𝐴 𝑤 ∈ 𝐵)
25		eluni 4879	. . . . 5 ⊢ (𝑤 ∈ ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ↔ ∃𝑧(𝑤 ∈ 𝑧 ∧ 𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵}))
26		vex 3467	. . . . . . . 8 ⊢ 𝑧 ∈ V
27		eqeq1 2773	. . . . . . . . 9 ⊢ (𝑦 = 𝑧 → (𝑦 = 𝐵 ↔ 𝑧 = 𝐵))
28	27	rexbidv 3195	. . . . . . . 8 ⊢ (𝑦 = 𝑧 → (∃𝑥 ∈ 𝐴 𝑦 = 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵))
29	26, 28	elab 3647	. . . . . . 7 ⊢ (𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ↔ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵)
30	29	anbi2i 634	. . . . . 6 ⊢ ((𝑤 ∈ 𝑧 ∧ 𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵}) ↔ (𝑤 ∈ 𝑧 ∧ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵))
31	30	exbii 1875	. . . . 5 ⊢ (∃𝑧(𝑤 ∈ 𝑧 ∧ 𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵}) ↔ ∃𝑧(𝑤 ∈ 𝑧 ∧ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵))
32	25, 31	bitri 278	. . . 4 ⊢ (𝑤 ∈ ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ↔ ∃𝑧(𝑤 ∈ 𝑧 ∧ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵))
33	20, 24, 32	3bitr4g 317	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → (𝑤 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵} ↔ 𝑤 ∈ ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵}))
34	33	eqrdv 2767	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵} = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵})
35	1, 34	eqtrid 2816	1 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵})

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567  ∃wex 1806   ∈ wcel 2149  {cab 2747  ∀wral 3085  ∃wrex 3095  ∪ cuni 4876  ∪ ciun 4960

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-11 2198  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rex 3096  df-v 3465  df-uni 4877  df-iun 4962

This theorem is referenced by:  dfiun2  5000  dfiun3g  5961  abnexg  7757  iunexg  7962  uniqs  8773  ac6num  10465  iunopn  23026  pnrmopn  23471  cncmp  23520  ptcmplem3  24182  iunmbl  25683  voliun  25684  sigaclcuni  34455  sigaclcu2  34457  sigaclci  34469  measvunilem  34549  meascnbl  34556  carsgclctunlem3  34657