Theorem dfiun2 2014

Description: Alternate definition of indexed union when B is a set. Definition 15(a) of [Suppes] p. 44.

Hypothesis

Ref	Expression
dfiun2.1	|- B e. V

Assertion

Ref	Expression
dfiun2	|- U.x e. A B = U.{y | E.x e. A y = B}

Distinct variable group(s):   x,y,A   y,B

Proof of Theorem dfiun2

Step	Hyp	Ref	Expression
1		df-rex 1206	. . . 4 |- (E.x e. A w e. B <-> E.x(x e. A /\ w e. B))
2		dfiun2.1	. . . . . . . . . 10 |- B e. V
3	2	clel3 1375	. . . . . . . . 9 |- (w e. B <-> E.z(z = B /\ w e. z))
4		exancom 736	. . . . . . . . 9 |- (E.z(z = B /\ w e. z) <-> E.z(w e. z /\ z = B))
5	3, 4	bitr 151	. . . . . . . 8 |- (w e. B <-> E.z(w e. z /\ z = B))
6	5	anbi2i 367	. . . . . . 7 |- ((x e. A /\ w e. B) <-> (x e. A /\ E.z(w e. z /\ z = B)))
7		19.42v 966	. . . . . . 7 |- (E.z(x e. A /\ (w e. z /\ z = B)) <-> (x e. A /\ E.z(w e. z /\ z = B)))
8	6, 7	bitr4 154	. . . . . 6 |- ((x e. A /\ w e. B) <-> E.z(x e. A /\ (w e. z /\ z = B)))
9	8	biex 733	. . . . 5 |- (E.x(x e. A /\ w e. B) <-> E.xE.z(x e. A /\ (w e. z /\ z = B)))
10		excom 728	. . . . 5 |- (E.xE.z(x e. A /\ (w e. z /\ z = B)) <-> E.zE.x(x e. A /\ (w e. z /\ z = B)))
11	9, 10	bitr 151	. . . 4 |- (E.x(x e. A /\ w e. B) <-> E.zE.x(x e. A /\ (w e. z /\ z = B)))
12		19.42v 966	. . . . . 6 |- (E.x(w e. z /\ (x e. A /\ z = B)) <-> (w e. z /\ E.x(x e. A /\ z = B)))
13		an12 370	. . . . . . 7 |- ((x e. A /\ (w e. z /\ z = B)) <-> (w e. z /\ (x e. A /\ z = B)))
14	13	biex 733	. . . . . 6 |- (E.x(x e. A /\ (w e. z /\ z = B)) <-> E.x(w e. z /\ (x e. A /\ z = B)))
15		visset 1350	. . . . . . . . 9 |- z e. V
16		cleq1 1107	. . . . . . . . . 10 |- (y = z -> (y = B <-> z = B))
17	16	birexdv 1220	. . . . . . . . 9 |- (y = z -> (E.x e. A y = B <-> E.x e. A z = B))
18	15, 17	elab 1415	. . . . . . . 8 |- (z e. {y | E.x e. A y = B} <-> E.x e. A z = B)
19		df-rex 1206	. . . . . . . 8 |- (E.x e. A z = B <-> E.x(x e. A /\ z = B))
20	18, 19	bitr 151	. . . . . . 7 |- (z e. {y | E.x e. A y = B} <-> E.x(x e. A /\ z = B))
21	20	anbi2i 367	. . . . . 6 |- ((w e. z /\ z e. {y | E.x e. A y = B}) <-> (w e. z /\ E.x(x e. A /\ z = B)))
22	12, 14, 21	3bitr4 158	. . . . 5 |- (E.x(x e. A /\ (w e. z /\ z = B)) <-> (w e. z /\ z e. {y | E.x e. A y = B}))
23	22	biex 733	. . . 4 |- (E.zE.x(x e. A /\ (w e. z /\ z = B)) <-> E.z(w e. z /\ z e. {y | E.x e. A y = B}))
24	1, 11, 23	3bitr 155	. . 3 |- (E.x e. A w e. B <-> E.z(w e. z /\ z e. {y | E.x e. A y = B}))
25	24	biabi 1181	. 2 |- {w | E.x e. A w e. B} = {w | E.z(w e. z /\ z e. {y | E.x e. A y = B})}
26		df-iun 1996	. 2 |- U.x e. A B = {w | E.x e. A w e. B}
27		df-uni 1920	. 2 |- U.{y | E.x e. A y = B} = {w | E.z(w e. z /\ z e. {y | E.x e. A y = B})}
28	25, 26, 27	3eqtr4 1126	1 |- U.x e. A B = U.{y | E.x e. A y = B}

Syntax hints:   /\ wa 196  E.wex 678   = weq 797   e. wel 803  {cab 1090   = wceq 1091   e. wcel 1092  E.wrex 1202  Vcvv 1348  U.cuni 1919  U.ciun 1994

This theorem is referenced by:  funcnvuni 2706  fniunfv 2860  iunex 2914  iunon 2947  rdglim2a 2988  kmlem10 3589  cardiun 3665

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  df-v 1349  df-uni 1920  df-iun 1996

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