Theorem dfiin2 2015

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

Hypothesis

Ref	Expression
dfiun2.1	|- B e. V

Assertion

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

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

Proof of Theorem dfiin2

Step	Hyp	Ref	Expression
1		df-ral 1205	. . . 4 |- (A.x e. A w e. B <-> A.x(x e. A -> w e. B))
2		dfiun2.1	. . . . . . . . 9 |- B e. V
3	2	clel4 1376	. . . . . . . 8 |- (w e. B <-> A.z(z = B -> w e. z))
4	3	imbi2i 160	. . . . . . 7 |- ((x e. A -> w e. B) <-> (x e. A -> A.z(z = B -> w e. z)))
5		19.21v 942	. . . . . . 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 154	. . . . . 6 |- ((x e. A -> w e. B) <-> A.z(x e. A -> (z = B -> w e. z)))
7	6	bial 695	. . . . 5 |- (A.x(x e. A -> w e. B) <-> A.xA.z(x e. A -> (z = B -> w e. z)))
8		alcom 715	. . . . 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 151	. . . 4 |- (A.x(x e. A -> w e. B) <-> A.zA.x(x e. A -> (z = B -> w e. z)))
10		impexp 276	. . . . . . . 8 |- (((x e. A /\ z = B) -> w e. z) <-> (x e. A -> (z = B -> w e. z)))
11	10	bial 695	. . . . . . 7 |- (A.x((x e. A /\ z = B) -> w e. z) <-> A.x(x e. A -> (z = B -> w e. z)))
12		19.23v 950	. . . . . . 7 |- (A.x((x e. A /\ z = B) -> w e. z) <-> (E.x(x e. A /\ z = B) -> w e. z))
13	11, 12	bitr3 153	. . . . . 6 |- (A.x(x e. A -> (z = B -> w e. z)) <-> (E.x(x e. A /\ z = B) -> w e. z))
14		visset 1350	. . . . . . . . 9 |- z e. V
15		cleq1 1107	. . . . . . . . . 10 |- (y = z -> (y = B <-> z = B))
16	15	birexdv 1220	. . . . . . . . 9 |- (y = z -> (E.x e. A y = B <-> E.x e. A z = B))
17	14, 16	elab 1415	. . . . . . . 8 |- (z e. {y | E.x e. A y = B} <-> E.x e. A z = B)
18		df-rex 1206	. . . . . . . 8 |- (E.x e. A z = B <-> E.x(x e. A /\ z = B))
19	17, 18	bitr 151	. . . . . . 7 |- (z e. {y | E.x e. A y = B} <-> E.x(x e. A /\ z = B))
20	19	imbi1i 161	. . . . . 6 |- ((z e. {y | E.x e. A y = B} -> w e. z) <-> (E.x(x e. A /\ z = B) -> w e. z))
21	13, 20	bitr4 154	. . . . 5 |- (A.x(x e. A -> (z = B -> w e. z)) <-> (z e. {y | E.x e. A y = B} -> w e. z))
22	21	bial 695	. . . 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))
23	1, 9, 22	3bitr 155	. . 3 |- (A.x e. A w e. B <-> A.z(z e. {y | E.x e. A y = B} -> w e. z))
24	23	biabi 1181	. 2 |- {w | A.x e. A w e. B} = {w | A.z(z e. {y | E.x e. A y = B} -> w e. z)}
25		df-iin 1997	. 2 |- |^|x e. A B = {w | A.x e. A w e. B}
26		df-int 1966	. 2 |- |^|{y | E.x e. A y = B} = {w | A.z(z e. {y | E.x e. A y = B} -> w e. z)}
27	24, 25, 26	3eqtr4 1126	1 |- |^|x e. A B = |^|{y | E.x e. A y = B}

Syntax hints:   -> wi 2   /\ wa 196  A.wal 672  E.wex 678   = weq 797   e. wel 803  {cab 1090   = wceq 1091   e. wcel 1092  A.wral 1201  E.wrex 1202  Vcvv 1348  |^|cint 1965  |^|ciin 1995

This theorem is referenced by:  iinon 2948  scott0 3542

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-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-int 1966  df-iin 1997

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