Theorem dfiun2g 3663

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 2333	. . . . . 6 ⊢ Ⅎx∀x ∈ A B ∈ 𝐶
2		rsp 2347	. . . . . . . 8 ⊢ (∀x ∈ A B ∈ 𝐶 → (x ∈ A → B ∈ 𝐶))
3		clel3g 2655	. . . . . . . 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 2299	. . . . 5 ⊢ (∀x ∈ A B ∈ 𝐶 → (∃x ∈ A z ∈ B ↔ ∃x ∈ A ∃y(y = B ∧ z ∈ y)))
7		rexcom4 2554	. . . . 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 2444	. . . . . 6 ⊢ (∃x ∈ A (y = B ∧ z ∈ y) ↔ (∃x ∈ A y = B ∧ z ∈ y))
10	9	exbii 1478	. . . . 5 ⊢ (∃y∃x ∈ A (y = B ∧ z ∈ y) ↔ ∃y(∃x ∈ A y = B ∧ z ∈ y))
11		exancom 1481	. . . . 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 3635	. . 3 ⊢ (z ∈ ∪ x ∈ A B ↔ ∃x ∈ A z ∈ B)
15		eluniab 3566	. . 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 2020	1 ⊢ (∀x ∈ A B ∈ 𝐶 → ∪ x ∈ A B = ∪ {y ∣ ∃x ∈ A y = B})

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1228  ∃wex 1362   ∈ wcel 1374  {cab 2008  ∀wral 2284  ∃wrex 2285  ∪ cuni 3554  ∪ ciun 3631

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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004

This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-uni 3555  df-iun 3633

This theorem is referenced by:  dfiun2  3665  dfiun3g  4516  fniunfv  5326  iunexg  5669  uniqs  6075