Theorem dfiin2g 3286

Description: Alternate definition of indexed intersection when B is a set. (Contributed by Jeff Hankins, 27-Aug-2009.)

Assertion

Ref	Expression
dfiin2g	|- (A.x e. A B e. C -> |^|_x e. A B = |^|{y | E.x e. A y = B})

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

Proof of Theorem dfiin2g

Step	Hyp	Ref	Expression
1		df-ral 2109	. . . . . 6 |- (A.x e. A B e. C <-> A.x(x e. A -> B e. C))
2		eleq2 1958	. . . . . . . . . . . . 13 |- (z = B -> (w e. z <-> w e. B))
3	2	biimprcd 173	. . . . . . . . . . . 12 |- (w e. B -> (z = B -> w e. z))
4	3	19.21aiv 1664	. . . . . . . . . . 11 |- (w e. B -> A.z(z = B -> w e. z))
5		eqid 1884	. . . . . . . . . . . 12 |- B = B
6		eqeq1 1890	. . . . . . . . . . . . . 14 |- (z = B -> (z = B <-> B = B))
7	6, 2	imbi12d 688	. . . . . . . . . . . . 13 |- (z = B -> ((z = B -> w e. z) <-> (B = B -> w e. B)))
8	7	cla4gv 2364	. . . . . . . . . . . 12 |- (B e. C -> (A.z(z = B -> w e. z) -> (B = B -> w e. B)))
9	5, 8	mpii 56	. . . . . . . . . . 11 |- (B e. C -> (A.z(z = B -> w e. z) -> w e. B))
10	4, 9	impbid2 576	. . . . . . . . . 10 |- (B e. C -> (w e. B <-> A.z(z = B -> w e. z)))
11	10	imim2i 11	. . . . . . . . 9 |- ((x e. A -> B e. C) -> (x e. A -> (w e. B <-> A.z(z = B -> w e. z))))
12	11	pm5.74d 645	. . . . . . . 8 |- ((x e. A -> B e. C) -> ((x e. A -> w e. B) <-> (x e. A -> A.z(z = B -> w e. z))))
13	12	alimi 1338	. . . . . . 7 |- (A.x(x e. A -> B e. C) -> A.x((x e. A -> w e. B) <-> (x e. A -> A.z(z = B -> w e. z))))
14		albi 1344	. . . . . . 7 |- (A.x((x e. A -> w e. B) <-> (x e. A -> A.z(z = B -> w e. z))) -> (A.x(x e. A -> w e. B) <-> A.x(x e. A -> A.z(z = B -> w e. z))))
15	13, 14	syl 12	. . . . . 6 |- (A.x(x e. A -> B e. C) -> (A.x(x e. A -> w e. B) <-> A.x(x e. A -> A.z(z = B -> w e. z))))
16	1, 15	sylbi 216	. . . . 5 |- (A.x e. A B e. C -> (A.x(x e. A -> w e. B) <-> A.x(x e. A -> A.z(z = B -> w e. z))))
17		df-ral 2109	. . . . . . . 8 |- (A.x e. A (z = B -> w e. z) <-> A.x(x e. A -> (z = B -> w e. z)))
18	17	albii 1346	. . . . . . 7 |- (A.zA.x e. A (z = B -> w e. z) <-> A.zA.x(x e. A -> (z = B -> w e. z)))
19		alcom 1379	. . . . . . 7 |- (A.xA.z(x e. A -> (z = B -> w e. z)) <-> A.zA.x(x e. A -> (z = B -> w e. z)))
20		19.21v 1663	. . . . . . . 8 |- (A.z(x e. A -> (z = B -> w e. z)) <-> (x e. A -> A.z(z = B -> w e. z)))
21	20	albii 1346	. . . . . . 7 |- (A.xA.z(x e. A -> (z = B -> w e. z)) <-> A.x(x e. A -> A.z(z = B -> w e. z)))
22	18, 19, 21	3bitr2ri 197	. . . . . 6 |- (A.x(x e. A -> A.z(z = B -> w e. z)) <-> A.zA.x e. A (z = B -> w e. z))
23		r19.23v 2208	. . . . . . . 8 |- (A.x e. A (z = B -> w e. z) <-> (E.x e. A z = B -> w e. z))
24		visset 2295	. . . . . . . . . 10 |- z e. _V
25		eqeq1 1890	. . . . . . . . . . 11 |- (y = z -> (y = B <-> z = B))
26	25	rexbidv 2124	. . . . . . . . . 10 |- (y = z -> (E.x e. A y = B <-> E.x e. A z = B))
27	24, 26	elab 2403	. . . . . . . . 9 |- (z e. {y | E.x e. A y = B} <-> E.x e. A z = B)
28	27	imbi1i 203	. . . . . . . 8 |- ((z e. {y | E.x e. A y = B} -> w e. z) <-> (E.x e. A z = B -> w e. z))
29	23, 28	bitr4i 193	. . . . . . 7 |- (A.x e. A (z = B -> w e. z) <-> (z e. {y | E.x e. A y = B} -> w e. z))
30	29	albii 1346	. . . . . 6 |- (A.zA.x e. A (z = B -> w e. z) <-> A.z(z e. {y | E.x e. A y = B} -> w e. z))
31	22, 30	bitri 190	. . . . 5 |- (A.x(x e. A -> A.z(z = B -> w e. z)) <-> A.z(z e. {y | E.x e. A y = B} -> w e. z))
32	16, 31	syl6bb 595	. . . 4 |- (A.x e. A B e. C -> (A.x(x e. A -> w e. B) <-> A.z(z e. {y | E.x e. A y = B} -> w e. z)))
33		df-ral 2109	. . . 4 |- (A.x e. A w e. B <-> A.x(x e. A -> w e. B))
34	32, 33	syl5bb 591	. . 3 |- (A.x e. A B e. C -> (A.x e. A w e. B <-> A.z(z e. {y | E.x e. A y = B} -> w e. z)))
35	34	abbidv 2008	. 2 |- (A.x e. A B e. C -> {w | A.x e. A w e. B} = {w | A.z(z e. {y | E.x e. A y = B} -> w e. z)})
36		df-iin 3258	. 2 |- |^|_x e. A B = {w | A.x e. A w e. B}
37		df-int 3215	. 2 |- |^|{y | E.x e. A y = B} = {w | A.z(z e. {y | E.x e. A y = B} -> w e. z)}
38	35, 36, 37	3eqtr4g 1953	1 |- (A.x e. A B e. C -> |^|_x e. A B = |^|{y | E.x e. A y = B})

Syntax hints:   -> wi 3   <-> wb 163  A.wal 1296   = wceq 1298   e. wcel 1300  {cab 1871  A.wral 2105  E.wrex 2106  |^|cint 3214  |^|_ciin 3256

This theorem is referenced by:  cexint2 14862  compfipin0lem 15435  compfipin0 15436

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ral 2109  df-rex 2110  df-v 2294  df-int 3215  df-iin 3258

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