Theorem dfiun2g 4000

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 2665	. . . . . 6 ⊢ Ⅎx∀x ∈ A B ∈ C
2		rsp 2675	. . . . . . . 8 ⊢ (∀x ∈ A B ∈ C → (x ∈ A → B ∈ C))
3		clel3g 2977	. . . . . . . 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 418	. . . . . 6 ⊢ ((∀x ∈ A B ∈ C ∧ x ∈ A) → (z ∈ B ↔ ∃y(y = B ∧ z ∈ y)))
6	1, 5	rexbida 2630	. . . . 5 ⊢ (∀x ∈ A B ∈ C → (∃x ∈ A z ∈ B ↔ ∃x ∈ A ∃y(y = B ∧ z ∈ y)))
7		rexcom4 2879	. . . . 5 ⊢ (∃x ∈ A ∃y(y = B ∧ z ∈ y) ↔ ∃y∃x ∈ A (y = B ∧ z ∈ y))
8	6, 7	syl6bb 252	. . . 4 ⊢ (∀x ∈ A B ∈ C → (∃x ∈ A z ∈ B ↔ ∃y∃x ∈ A (y = B ∧ z ∈ y)))
9		r19.41v 2765	. . . . . 6 ⊢ (∃x ∈ A (y = B ∧ z ∈ y) ↔ (∃x ∈ A y = B ∧ z ∈ y))
10	9	exbii 1582	. . . . 5 ⊢ (∃y∃x ∈ A (y = B ∧ z ∈ y) ↔ ∃y(∃x ∈ A y = B ∧ z ∈ y))
11		exancom 1586	. . . . 5 ⊢ (∃y(∃x ∈ A y = B ∧ z ∈ y) ↔ ∃y(z ∈ y ∧ ∃x ∈ A y = B))
12	10, 11	bitri 240	. . . 4 ⊢ (∃y∃x ∈ A (y = B ∧ z ∈ y) ↔ ∃y(z ∈ y ∧ ∃x ∈ A y = B))
13	8, 12	syl6bb 252	. . 3 ⊢ (∀x ∈ A B ∈ C → (∃x ∈ A z ∈ B ↔ ∃y(z ∈ y ∧ ∃x ∈ A y = B)))
14		eliun 3974	. . 3 ⊢ (z ∈ ∪x ∈ A B ↔ ∃x ∈ A z ∈ B)
15		eluniab 3904	. . 3 ⊢ (z ∈ ∪{y ∣ ∃x ∈ A y = B} ↔ ∃y(z ∈ y ∧ ∃x ∈ A y = B))
16	13, 14, 15	3bitr4g 279	. 2 ⊢ (∀x ∈ A B ∈ C → (z ∈ ∪x ∈ A B ↔ z ∈ ∪{y ∣ ∃x ∈ A y = B}))
17	16	eqrdv 2351	1 ⊢ (∀x ∈ A B ∈ C → ∪x ∈ A B = ∪{y ∣ ∃x ∈ A y = B})

This proof depends on syntax axioms:   → wi 4   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  {cab 2339  ∀wral 2615  ∃wrex 2616  ∪cuni 3892  ∪ciun 3970

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This proof depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620  df-rex 2621  df-v 2862  df-uni 3893  df-iun 3972

This theorem is used by:  dfiun2  4002  uniqs  5985