Theorem xpiindi 5143

Description: Distributive law for Cartesian product over indexed intersection. (Contributed by Mario Carneiro, 21-Mar-2015.)

Assertion

Ref	Expression
xpiindi

Distinct variable groups:   ,   ,

Proof of Theorem xpiindi

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

Step	Hyp	Ref	Expression
1		relxp 5115	. . . . . 6
2	1	rgenw 2818	. . . . 5
3		r19.2z 3918	. . . . 5
4	2, 3	mpan2 671	. . . 4
5		reliin 5129	. . . 4
6	4, 5	syl 16	. . 3
7		relxp 5115	. . 3
8	6, 7	jctil 537	. 2
9		r19.28zv 3924	. . . . . 6
10	9	bicomd 201	. . . . 5
11		vex 3112	. . . . . . 7
12		eliin 4336	. . . . . . 7
13	11, 12	ax-mp 5	. . . . . 6
14	13	anbi2i 694	. . . . 5
15		opelxp 5034	. . . . . 6
16	15	ralbii 2888	. . . . 5
17	10, 14, 16	3bitr4g 288	. . . 4
18		opelxp 5034	. . . 4
19		opex 4716	. . . . 5
20		eliin 4336	. . . . 5
21	19, 20	ax-mp 5	. . . 4
22	17, 18, 21	3bitr4g 288	. . 3
23	22	eqrelrdv2 5107	. 2
24	8, 23	mpancom 669	1

Syntax hints:  ->wi 4  <->wb 184  /\wa 369  =wceq 1395  e.wcel 1818  =/=wne 2652  A.wral 2807  E.wrex 2808  cvv 3109  c0 3784  <.cop 4035  |^|_ciin 4331  X.cxp 5002  Relwrel 5009

This theorem is referenced by:  xpriindi  5144

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-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  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-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-iin 4333  df-opab 4511  df-xp 5010  df-rel 5011