Theorem dfiun2g 4552

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.)

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 variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfra1 2941	. . . . . 6 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶
2		rsp 2929	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → (𝑥 ∈ 𝐴 → 𝐵 ∈ 𝐶))
3		clel3g 3340	. . . . . . . 8 ⊢ (𝐵 ∈ 𝐶 → (𝑧 ∈ 𝐵 ↔ ∃𝑦(𝑦 = 𝐵 ∧ 𝑧 ∈ 𝑦)))
4	2, 3	syl6 35	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → (𝑥 ∈ 𝐴 → (𝑧 ∈ 𝐵 ↔ ∃𝑦(𝑦 = 𝐵 ∧ 𝑧 ∈ 𝑦))))
5	4	imp 445	. . . . . 6 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 ∧ 𝑥 ∈ 𝐴) → (𝑧 ∈ 𝐵 ↔ ∃𝑦(𝑦 = 𝐵 ∧ 𝑧 ∈ 𝑦)))
6	1, 5	rexbida 3047	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → (∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦(𝑦 = 𝐵 ∧ 𝑧 ∈ 𝑦)))
7		rexcom4 3225	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦(𝑦 = 𝐵 ∧ 𝑧 ∈ 𝑦) ↔ ∃𝑦∃𝑥 ∈ 𝐴 (𝑦 = 𝐵 ∧ 𝑧 ∈ 𝑦))
8	6, 7	syl6bb 276	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → (∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ↔ ∃𝑦∃𝑥 ∈ 𝐴 (𝑦 = 𝐵 ∧ 𝑧 ∈ 𝑦)))
9		r19.41v 3089	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐴 (𝑦 = 𝐵 ∧ 𝑧 ∈ 𝑦) ↔ (∃𝑥 ∈ 𝐴 𝑦 = 𝐵 ∧ 𝑧 ∈ 𝑦))
10	9	exbii 1774	. . . . 5 ⊢ (∃𝑦∃𝑥 ∈ 𝐴 (𝑦 = 𝐵 ∧ 𝑧 ∈ 𝑦) ↔ ∃𝑦(∃𝑥 ∈ 𝐴 𝑦 = 𝐵 ∧ 𝑧 ∈ 𝑦))
11		exancom 1787	. . . . 5 ⊢ (∃𝑦(∃𝑥 ∈ 𝐴 𝑦 = 𝐵 ∧ 𝑧 ∈ 𝑦) ↔ ∃𝑦(𝑧 ∈ 𝑦 ∧ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵))
12	10, 11	bitri 264	. . . 4 ⊢ (∃𝑦∃𝑥 ∈ 𝐴 (𝑦 = 𝐵 ∧ 𝑧 ∈ 𝑦) ↔ ∃𝑦(𝑧 ∈ 𝑦 ∧ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵))
13	8, 12	syl6bb 276	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → (∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ↔ ∃𝑦(𝑧 ∈ 𝑦 ∧ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵)))
14		eliun 4524	. . 3 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵)
15		eluniab 4447	. . 3 ⊢ (𝑧 ∈ ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ↔ ∃𝑦(𝑧 ∈ 𝑦 ∧ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵))
16	13, 14, 15	3bitr4g 303	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ 𝑧 ∈ ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵}))
17	16	eqrdv 2620	1 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵})

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483  ∃wex 1704   ∈ wcel 1990  {cab 2608  ∀wral 2912  ∃wrex 2913  ∪ cuni 4436  ∪ ciun 4520

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-v 3202  df-uni 4437  df-iun 4522

This theorem is referenced by:  dfiun2  4554  dfiun3g  5378  abnexg  6964  iunexg  7143  uniqs  7807  ac6num  9301  iunopn  20703  pnrmopn  21147  cncmp  21195  ptcmplem3  21858  iunmbl  23321  voliun  23322  sigaclcuni  30181  sigaclcu2  30183  sigaclci  30195  measvunilem  30275  meascnbl  30282  carsgclctunlem3  30382  uniqsALTV  34101