Theorem dfiun2g 3898

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 2497	. . . . . 6 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶
2		rsp 2513	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → (𝑥 ∈ 𝐴 → 𝐵 ∈ 𝐶))
3		clel3g 2860	. . . . . . . 8 ⊢ (𝐵 ∈ 𝐶 → (𝑧 ∈ 𝐵 ↔ ∃𝑦(𝑦 = 𝐵 ∧ 𝑧 ∈ 𝑦)))
4	2, 3	syl6 33	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → (𝑥 ∈ 𝐴 → (𝑧 ∈ 𝐵 ↔ ∃𝑦(𝑦 = 𝐵 ∧ 𝑧 ∈ 𝑦))))
5	4	imp 123	. . . . . 6 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 ∧ 𝑥 ∈ 𝐴) → (𝑧 ∈ 𝐵 ↔ ∃𝑦(𝑦 = 𝐵 ∧ 𝑧 ∈ 𝑦)))
6	1, 5	rexbida 2461	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → (∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦(𝑦 = 𝐵 ∧ 𝑧 ∈ 𝑦)))
7		rexcom4 2749	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦(𝑦 = 𝐵 ∧ 𝑧 ∈ 𝑦) ↔ ∃𝑦∃𝑥 ∈ 𝐴 (𝑦 = 𝐵 ∧ 𝑧 ∈ 𝑦))
8	6, 7	bitrdi 195	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → (∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ↔ ∃𝑦∃𝑥 ∈ 𝐴 (𝑦 = 𝐵 ∧ 𝑧 ∈ 𝑦)))
9		r19.41v 2622	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐴 (𝑦 = 𝐵 ∧ 𝑧 ∈ 𝑦) ↔ (∃𝑥 ∈ 𝐴 𝑦 = 𝐵 ∧ 𝑧 ∈ 𝑦))
10	9	exbii 1593	. . . . 5 ⊢ (∃𝑦∃𝑥 ∈ 𝐴 (𝑦 = 𝐵 ∧ 𝑧 ∈ 𝑦) ↔ ∃𝑦(∃𝑥 ∈ 𝐴 𝑦 = 𝐵 ∧ 𝑧 ∈ 𝑦))
11		exancom 1596	. . . . 5 ⊢ (∃𝑦(∃𝑥 ∈ 𝐴 𝑦 = 𝐵 ∧ 𝑧 ∈ 𝑦) ↔ ∃𝑦(𝑧 ∈ 𝑦 ∧ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵))
12	10, 11	bitri 183	. . . 4 ⊢ (∃𝑦∃𝑥 ∈ 𝐴 (𝑦 = 𝐵 ∧ 𝑧 ∈ 𝑦) ↔ ∃𝑦(𝑧 ∈ 𝑦 ∧ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵))
13	8, 12	bitrdi 195	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → (∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ↔ ∃𝑦(𝑧 ∈ 𝑦 ∧ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵)))
14		eliun 3870	. . 3 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵)
15		eluniab 3801	. . 3 ⊢ (𝑧 ∈ ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ↔ ∃𝑦(𝑧 ∈ 𝑦 ∧ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵))
16	13, 14, 15	3bitr4g 222	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ 𝑧 ∈ ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵}))
17	16	eqrdv 2163	1 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵})

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1343  ∃wex 1480   ∈ wcel 2136  {cab 2151  ∀wral 2444  ∃wrex 2445  ∪ cuni 3789  ∪ ciun 3866

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-uni 3790  df-iun 3868

This theorem is referenced by:  dfiun2  3900  abnexg  4424  dfiun3g  4861  fniunfv  5730  iunexg  6087  uniqs  6559  iunopn  12640