Theorem dfiin2g 4363

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

Assertion

Ref	Expression
dfiin2g

Distinct variable groups:   ,   ,   ,

Proof of Theorem dfiin2g

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

Step	Hyp	Ref	Expression
1		df-ral 2812	. . . 4
2		df-ral 2812	. . . . . 6
3		eleq2 2530	. . . . . . . . . . . . 13
4	3	biimprcd 225	. . . . . . . . . . . 12
5	4	alrimiv 1719	. . . . . . . . . . 11
6		eqid 2457	. . . . . . . . . . . 12
7		eqeq1 2461	. . . . . . . . . . . . . 14
8	7, 3	imbi12d 320	. . . . . . . . . . . . 13
9	8	spcgv 3194	. . . . . . . . . . . 12
10	6, 9	mpii 43	. . . . . . . . . . 11
11	5, 10	impbid2 204	. . . . . . . . . 10
12	11	imim2i 14	. . . . . . . . 9
13	12	pm5.74d 247	. . . . . . . 8
14	13	alimi 1633	. . . . . . 7
15		albi 1639	. . . . . . 7
16	14, 15	syl 16	. . . . . 6
17	2, 16	sylbi 195	. . . . 5
18		df-ral 2812	. . . . . . . 8
19	18	albii 1640	. . . . . . 7
20		alcom 1845	. . . . . . 7
21	19, 20	bitr4i 252	. . . . . 6
22		r19.23v 2937	. . . . . . . 8
23		vex 3112	. . . . . . . . . 10
24		eqeq1 2461	. . . . . . . . . . 11
25	24	rexbidv 2968	. . . . . . . . . 10
26	23, 25	elab 3246	. . . . . . . . 9
27	26	imbi1i 325	. . . . . . . 8
28	22, 27	bitr4i 252	. . . . . . 7
29	28	albii 1640	. . . . . 6
30		19.21v 1729	. . . . . . 7
31	30	albii 1640	. . . . . 6
32	21, 29, 31	3bitr3ri 276	. . . . 5
33	17, 32	syl6bb 261	. . . 4
34	1, 33	syl5bb 257	. . 3
35	34	abbidv 2593	. 2
36		df-iin 4333	. 2
37		df-int 4287	. 2
38	35, 36, 37	3eqtr4g 2523	1

Syntax hints:  ->wi 4  <->wb 184  A.wal 1393  =wceq 1395  e.wcel 1818  {cab 2442  A.wral 2807  E.wrex 2808  |^|cint 4286  |^|_ciin 4331

This theorem is referenced by:  dfiin2  4365  iinexg  4612  dfiin3g  5261  iinfi  7897  mreiincl  14993  iinopn  19411  clsval2  19551  alexsublem  20544

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-rex 2813  df-v 3111  df-int 4287  df-iin 4333