Theorem dfiun2g 3680

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 2349	. . . . . 6 ⊢ Ⅎx∀x ∈ A B ∈ 𝐶
2		rsp 2363	. . . . . . . 8 ⊢ (∀x ∈ A B ∈ 𝐶 → (x ∈ A → B ∈ 𝐶))
3		clel3g 2672	. . . . . . . 8 ⊢ (B ∈ 𝐶 → (z ∈ B ↔ ∃y(y = B ∧ z ∈ y)))
4	2, 3	syl6 29	. . . . . . 7 ⊢ (∀x ∈ A B ∈ 𝐶 → (x ∈ A → (z ∈ B ↔ ∃y(y = B ∧ z ∈ y))))
5	4	imp 115	. . . . . 6 ⊢ ((∀x ∈ A B ∈ 𝐶 ∧ x ∈ A) → (z ∈ B ↔ ∃y(y = B ∧ z ∈ y)))
6	1, 5	rexbida 2315	. . . . 5 ⊢ (∀x ∈ A B ∈ 𝐶 → (∃x ∈ A z ∈ B ↔ ∃x ∈ A ∃y(y = B ∧ z ∈ y)))
7		rexcom4 2571	. . . . 5 ⊢ (∃x ∈ A ∃y(y = B ∧ z ∈ y) ↔ ∃y∃x ∈ A (y = B ∧ z ∈ y))
8	6, 7	syl6bb 185	. . . 4 ⊢ (∀x ∈ A B ∈ 𝐶 → (∃x ∈ A z ∈ B ↔ ∃y∃x ∈ A (y = B ∧ z ∈ y)))
9		r19.41v 2460	. . . . . 6 ⊢ (∃x ∈ A (y = B ∧ z ∈ y) ↔ (∃x ∈ A y = B ∧ z ∈ y))
10	9	exbii 1493	. . . . 5 ⊢ (∃y∃x ∈ A (y = B ∧ z ∈ y) ↔ ∃y(∃x ∈ A y = B ∧ z ∈ y))
11		exancom 1496	. . . . 5 ⊢ (∃y(∃x ∈ A y = B ∧ z ∈ y) ↔ ∃y(z ∈ y ∧ ∃x ∈ A y = B))
12	10, 11	bitri 173	. . . 4 ⊢ (∃y∃x ∈ A (y = B ∧ z ∈ y) ↔ ∃y(z ∈ y ∧ ∃x ∈ A y = B))
13	8, 12	syl6bb 185	. . 3 ⊢ (∀x ∈ A B ∈ 𝐶 → (∃x ∈ A z ∈ B ↔ ∃y(z ∈ y ∧ ∃x ∈ A y = B)))
14		eliun 3652	. . 3 ⊢ (z ∈ ∪ x ∈ A B ↔ ∃x ∈ A z ∈ B)
15		eluniab 3583	. . 3 ⊢ (z ∈ ∪ {y ∣ ∃x ∈ A y = B} ↔ ∃y(z ∈ y ∧ ∃x ∈ A y = B))
16	13, 14, 15	3bitr4g 212	. 2 ⊢ (∀x ∈ A B ∈ 𝐶 → (z ∈ ∪ x ∈ A B ↔ z ∈ ∪ {y ∣ ∃x ∈ A y = B}))
17	16	eqrdv 2035	1 ⊢ (∀x ∈ A B ∈ 𝐶 → ∪ x ∈ A B = ∪ {y ∣ ∃x ∈ A y = B})

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1242  ∃wex 1378   ∈ wcel 1390  {cab 2023  ∀wral 2300  ∃wrex 2301  ∪ cuni 3571  ∪ ciun 3648

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-uni 3572  df-iun 3650

This theorem is referenced by:  dfiun2  3682  dfiun3g  4532  fniunfv  5344  iunexg  5688  uniqs  6100