Theorem dfiin2g 3681

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 2305	. . . 4 |- (A.x e. A w e. B <-> A.x(x e. A -> w e. B))
2		df-ral 2305	. . . . . 6 |- (A.x e. A B e. C <-> A.x(x e. A -> B e. C))
3		eleq2 2098	. . . . . . . . . . . . 13 |- (z = B -> (w e. z <-> w e. B))
4	3	biimprcd 149	. . . . . . . . . . . 12 |- (w e. B -> (z = B -> w e. z))
5	4	alrimiv 1751	. . . . . . . . . . 11 |- (w e. B -> A.z(z = B -> w e. z))
6		eqid 2037	. . . . . . . . . . . 12 |- B = B
7		eqeq1 2043	. . . . . . . . . . . . . 14 |- (z = B -> (z = B <-> B = B))
8	7, 3	imbi12d 223	. . . . . . . . . . . . 13 |- (z = B -> ((z = B -> w e. z) <-> (B = B -> w e. B)))
9	8	spcgv 2634	. . . . . . . . . . . 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 131	. . . . . . . . . 10 |- (B e. C -> (w e. B <-> A.z(z = B -> w e. z)))
12	11	imim2i 12	. . . . . . . . 9 |- ((x e. A -> B e. C) -> (x e. A -> (w e. B <-> A.z(z = B -> w e. z))))
13	12	pm5.74d 171	. . . . . . . 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 1341	. . . . . . 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 1354	. . . . . . 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 14	. . . . . 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 114	. . . . 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 2305	. . . . . . . 8 |- (A.x e. A (z = B -> w e. z) <-> A.x(x e. A -> (z = B -> w e. z)))
19	18	albii 1356	. . . . . . 7 |- (A.zA.x e. A (z = B -> w e. z) <-> A.zA.x(x e. A -> (z = B -> w e. z)))
20		alcom 1364	. . . . . . 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 176	. . . . . 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 2419	. . . . . . . 8 |- (A.x e. A (z = B -> w e. z) <-> (E.x e. A z = B -> w e. z))
23		vex 2554	. . . . . . . . . 10 |- z e. _V
24		eqeq1 2043	. . . . . . . . . . 11 |- (y = z -> (y = B <-> z = B))
25	24	rexbidv 2321	. . . . . . . . . 10 |- (y = z -> (E.x e. A y = B <-> E.x e. A z = B))
26	23, 25	elab 2681	. . . . . . . . 9 |- (z e. {y | E.x e. A y = B } <-> E.x e. A z = B)
27	26	imbi1i 227	. . . . . . . 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 176	. . . . . . 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 1356	. . . . . 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 1750	. . . . . . 7 |- (A.z(x e. A -> (z = B -> w e. z)) <-> (x e. A -> A.z(z = B -> w e. z)))
31	30	albii 1356	. . . . . 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 200	. . . . 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 185	. . . 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 181	. . 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 2152	. 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 3651	. 2 |- |^|_x e. A B = {w | A.x e. A w e. B }
37		df-int 3607	. 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 2094	1 |- (A.x e. A B e. C -> |^|_x e. A B = |^| {y | E.x e. A y = B })

Syntax hints:   -> wi 4   <-> wb 98  A.wal 1240   = wceq 1242   e. wcel 1390   {cab 2023  A.wral 2300  E.wrex 2301   |^|cint 3606   |^|_ciin 3649

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-int 3607  df-iin 3651

This theorem is referenced by:  dfiin2  3683  iinexgm  3899  dfiin3g  4533  fniinfv  5174