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	|- (A.x e. A B e. C -> U_x e. A B = U.{y | E.x e. 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 |- F/xA.x e. A B e. C
2		rsp 2675	. . . . . . . 8 |- (A.x e. A B e. C -> (x e. A -> B e. C))
3		clel3g 2977	. . . . . . . 8 |- (B e. C -> (z e. B <-> E.y(y = B /\ z e. y)))
4	2, 3	syl6 29	. . . . . . 7 |- (A.x e. A B e. C -> (x e. A -> (z e. B <-> E.y(y = B /\ z e. y))))
5	4	imp 418	. . . . . 6 |- ((A.x e. A B e. C /\ x e. A) -> (z e. B <-> E.y(y = B /\ z e. y)))
6	1, 5	rexbida 2630	. . . . 5 |- (A.x e. A B e. C -> (E.x e. A z e. B <-> E.x e. A E.y(y = B /\ z e. y)))
7		rexcom4 2879	. . . . 5 |- (E.x e. A E.y(y = B /\ z e. y) <-> E.yE.x e. A (y = B /\ z e. y))
8	6, 7	syl6bb 252	. . . 4 |- (A.x e. A B e. C -> (E.x e. A z e. B <-> E.yE.x e. A (y = B /\ z e. y)))
9		r19.41v 2765	. . . . . 6 |- (E.x e. A (y = B /\ z e. y) <-> (E.x e. A y = B /\ z e. y))
10	9	exbii 1582	. . . . 5 |- (E.yE.x e. A (y = B /\ z e. y) <-> E.y(E.x e. A y = B /\ z e. y))
11		exancom 1586	. . . . 5 |- (E.y(E.x e. A y = B /\ z e. y) <-> E.y(z e. y /\ E.x e. A y = B))
12	10, 11	bitri 240	. . . 4 |- (E.yE.x e. A (y = B /\ z e. y) <-> E.y(z e. y /\ E.x e. A y = B))
13	8, 12	syl6bb 252	. . 3 |- (A.x e. A B e. C -> (E.x e. A z e. B <-> E.y(z e. y /\ E.x e. A y = B)))
14		eliun 3974	. . 3 |- (z e. U_x e. A B <-> E.x e. A z e. B)
15		eluniab 3904	. . 3 |- (z e. U.{y | E.x e. A y = B} <-> E.y(z e. y /\ E.x e. A y = B))
16	13, 14, 15	3bitr4g 279	. 2 |- (A.x e. A B e. C -> (z e. U_x e. A B <-> z e. U.{y | E.x e. A y = B}))
17	16	eqrdv 2351	1 |- (A.x e. A B e. C -> U_x e. A B = U.{y | E.x e. A y = B})

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358  E.wex 1541   = wceq 1642   e. wcel 1710  {cab 2339  A.wral 2615  E.wrex 2616  U.cuni 3892  U_ciun 3970

This theorem was proved from 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 theorem 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 referenced by:  dfiun2  4002  uniqs  5985