Theorem dfiin2g 4001

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

Allowed substitution hints:   A(x)   B(x)   C(x,y)

Proof of Theorem dfiin2g

Dummy variables z w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ral 2620	. . . 4 |- (A.x e. A w e. B <-> A.x(x e. A -> w e. B))
2		df-ral 2620	. . . . . 6 |- (A.x e. A B e. C <-> A.x(x e. A -> B e. C))
3		eleq2 2414	. . . . . . . . . . . . 13 |- (z = B -> (w e. z <-> w e. B))
4	3	biimprcd 216	. . . . . . . . . . . 12 |- (w e. B -> (z = B -> w e. z))
5	4	alrimiv 1631	. . . . . . . . . . 11 |- (w e. B -> A.z(z = B -> w e. z))
6		eqid 2353	. . . . . . . . . . . 12 |- B = B
7		eqeq1 2359	. . . . . . . . . . . . . 14 |- (z = B -> (z = B <-> B = B))
8	7, 3	imbi12d 311	. . . . . . . . . . . . 13 |- (z = B -> ((z = B -> w e. z) <-> (B = B -> w e. B)))
9	8	spcgv 2940	. . . . . . . . . . . 12 |- (B e. C -> (A.z(z = B -> w e. z) -> (B = B -> w e. B)))
10	6, 9	mpii 39	. . . . . . . . . . 11 |- (B e. C -> (A.z(z = B -> w e. z) -> w e. B))
11	5, 10	impbid2 195	. . . . . . . . . 10 |- (B e. C -> (w e. B <-> A.z(z = B -> w e. z)))
12	11	imim2i 13	. . . . . . . . 9 |- ((x e. A -> B e. C) -> (x e. A -> (w e. B <-> A.z(z = B -> w e. z))))
13	12	pm5.74d 238	. . . . . . . 8 |- ((x e. A -> B e. C) -> ((x e. A -> w e. B) <-> (x e. A -> A.z(z = B -> w e. z))))
14	13	alimi 1559	. . . . . . 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))))
15		albi 1564	. . . . . . 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))))
16	14, 15	syl 15	. . . . . 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))))
17	2, 16	sylbi 187	. . . . 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))))
18		df-ral 2620	. . . . . . . 8 |- (A.x e. A (z = B -> w e. z) <-> A.x(x e. A -> (z = B -> w e. z)))
19	18	albii 1566	. . . . . . 7 |- (A.zA.x e. A (z = B -> w e. z) <-> A.zA.x(x e. A -> (z = B -> w e. z)))
20		alcom 1737	. . . . . . 7 |- (A.xA.z(x e. A -> (z = B -> w e. z)) <-> A.zA.x(x e. A -> (z = B -> w e. z)))
21	19, 20	bitr4i 243	. . . . . 6 |- (A.zA.x e. A (z = B -> w e. z) <-> A.xA.z(x e. A -> (z = B -> w e. z)))
22		r19.23v 2731	. . . . . . . 8 |- (A.x e. A (z = B -> w e. z) <-> (E.x e. A z = B -> w e. z))
23		vex 2863	. . . . . . . . . 10 |- z e. _V
24		eqeq1 2359	. . . . . . . . . . 11 |- (y = z -> (y = B <-> z = B))
25	24	rexbidv 2636	. . . . . . . . . 10 |- (y = z -> (E.x e. A y = B <-> E.x e. A z = B))
26	23, 25	elab 2986	. . . . . . . . 9 |- (z e. {y | E.x e. A y = B} <-> E.x e. A z = B)
27	26	imbi1i 315	. . . . . . . 8 |- ((z e. {y | E.x e. A y = B} -> w e. z) <-> (E.x e. A z = B -> w e. z))
28	22, 27	bitr4i 243	. . . . . . 7 |- (A.x e. A (z = B -> w e. z) <-> (z e. {y | E.x e. A y = B} -> w e. z))
29	28	albii 1566	. . . . . 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))
30		19.21v 1890	. . . . . . 7 |- (A.z(x e. A -> (z = B -> w e. z)) <-> (x e. A -> A.z(z = B -> w e. z)))
31	30	albii 1566	. . . . . 6 |- (A.xA.z(x e. A -> (z = B -> w e. z)) <-> A.x(x e. A -> A.z(z = B -> w e. z)))
32	21, 29, 31	3bitr3ri 267	. . . . 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))
33	17, 32	syl6bb 252	. . . 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)))
34	1, 33	syl5bb 248	. . 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 2468	. 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 3973	. 2 |- |^|_x e. A B = {w | A.x e. A w e. B}
37		df-int 3928	. 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 2410	1 |- (A.x e. A B e. C -> |^|_x e. A B = |^|{y | E.x e. A y = B})

Syntax hints:   -> wi 4   <-> wb 176  A.wal 1540   = wceq 1642   e. wcel 1710  {cab 2339  A.wral 2615  E.wrex 2616  |^|cint 3927  |^|_ciin 3971

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-int 3928  df-iin 3973

This theorem is referenced by:  dfiin2  4003