Theorem dfiin2 2583

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

Hypothesis

Ref	Expression
dfiin2.1	|- B e. V

Assertion

Ref	Expression
dfiin2	|- |^|_x e. A B = |^|{y | E.x e. A y = B}

Distinct variable groups:   y,A   y,B   x,y

Proof of Theorem dfiin2

Step	Hyp	Ref	Expression
1		df-ral 1646	. . . 4 |- (A.x e. A w e. B <-> A.x(x e. A -> w e. B))
2		dfiin2.1	. . . . . . . . 9 |- B e. V
3	2	clel4 1890	. . . . . . . 8 |- (w e. B <-> A.z(z = B -> w e. z))
4	3	imbi2i 185	. . . . . . 7 |- ((x e. A -> w e. B) <-> (x e. A -> A.z(z = B -> w e. z)))
5		19.21v 1283	. . . . . . 7 |- (A.z(x e. A -> (z = B -> w e. z)) <-> (x e. A -> A.z(z = B -> w e. z)))
6	4, 5	bitr4 176	. . . . . 6 |- ((x e. A -> w e. B) <-> A.z(x e. A -> (z = B -> w e. z)))
7	6	albii 997	. . . . 5 |- (A.x(x e. A -> w e. B) <-> A.xA.z(x e. A -> (z = B -> w e. z)))
8		alcom 1030	. . . . 5 |- (A.xA.z(x e. A -> (z = B -> w e. z)) <-> A.zA.x(x e. A -> (z = B -> w e. z)))
9	7, 8	bitr 173	. . . 4 |- (A.x(x e. A -> w e. B) <-> A.zA.x(x e. A -> (z = B -> w e. z)))
10		visset 1809	. . . . . . . . 9 |- z e. V
11		eqeq1 1478	. . . . . . . . . 10 |- (y = z -> (y = B <-> z = B))
12	11	rexbidv 1661	. . . . . . . . 9 |- (y = z -> (E.x e. A y = B <-> E.x e. A z = B))
13	10, 12	elab 1893	. . . . . . . 8 |- (z e. {y | E.x e. A y = B} <-> E.x e. A z = B)
14		df-rex 1647	. . . . . . . 8 |- (E.x e. A z = B <-> E.x(x e. A /\ z = B))
15	13, 14	bitr 173	. . . . . . 7 |- (z e. {y | E.x e. A y = B} <-> E.x(x e. A /\ z = B))
16	15	imbi1i 186	. . . . . 6 |- ((z e. {y | E.x e. A y = B} -> w e. z) <-> (E.x(x e. A /\ z = B) -> w e. z))
17		19.23v 1291	. . . . . 6 |- (A.x((x e. A /\ z = B) -> w e. z) <-> (E.x(x e. A /\ z = B) -> w e. z))
18		impexp 347	. . . . . . 7 |- (((x e. A /\ z = B) -> w e. z) <-> (x e. A -> (z = B -> w e. z)))
19	18	albii 997	. . . . . 6 |- (A.x((x e. A /\ z = B) -> w e. z) <-> A.x(x e. A -> (z = B -> w e. z)))
20	16, 17, 19	3bitr2r 180	. . . . 5 |- (A.x(x e. A -> (z = B -> w e. z)) <-> (z e. {y | E.x e. A y = B} -> w e. z))
21	20	albii 997	. . . 4 |- (A.zA.x(x e. A -> (z = B -> w e. z)) <-> A.z(z e. {y | E.x e. A y = B} -> w e. z))
22	1, 9, 21	3bitr 177	. . 3 |- (A.x e. A w e. B <-> A.z(z e. {y | E.x e. A y = B} -> w e. z))
23	22	abbii 1572	. 2 |- {w | A.x e. A w e. B} = {w | A.z(z e. {y | E.x e. A y = B} -> w e. z)}
24		df-iin 2564	. 2 |- |^|_x e. A B = {w | A.x e. A w e. B}
25		df-int 2529	. 2 |- |^|{y | E.x e. A y = B} = {w | A.z(z e. {y | E.x e. A y = B} -> w e. z)}
26	23, 24, 25	3eqtr4 1502	1 |- |^|_x e. A B = |^|{y | E.x e. A y = B}

Syntax hints:   -> wi 3   /\ wa 223  A.wal 952   = wceq 954   e. wcel 956  E.wex 978  {cab 1461  A.wral 1642  E.wrex 1643  Vcvv 1807  |^|cint 2528  |^|_ciin 2562

This theorem is referenced by:  fniinfv 3757  iinon 3901  scott0 4697

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-ral 1646  df-rex 1647  df-v 1808  df-int 2529  df-iin 2564

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