Theorem xpiundir 4976

Description: Distributive law for Cartesian product over indexed union. (Contributed by Mario Carneiro, 27-Apr-2014.)

Assertion

Ref	Expression
xpiundir

Distinct variable group:   ,

Proof of Theorem xpiundir

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

Step	Hyp	Ref	Expression
1		rexcom4 3072	. . . . 5
2		df-rex 2798	. . . . . 6
3	2	rexbii 2813	. . . . 5
4		eliun 4257	. . . . . . . 8
5	4	anbi1i 695	. . . . . . 7
6		r19.41v 2953	. . . . . . 7
7	5, 6	bitr4i 252	. . . . . 6
8	7	exbii 1635	. . . . 5
9	1, 3, 8	3bitr4ri 278	. . . 4
10		df-rex 2798	. . . 4
11		elxp2 4940	. . . . 5
12	11	rexbii 2813	. . . 4
13	9, 10, 12	3bitr4i 277	. . 3
14		elxp2 4940	. . 3
15		eliun 4257	. . 3
16	13, 14, 15	3bitr4i 277	. 2
17	16	eqriv 2446	1

Syntax hints:  /\wa 369  =wceq 1370  E.wex 1587  e.wcel 1757  E.wrex 2793  <.cop 3965  U_ciun 4253  X.cxp 4920

This theorem is referenced by:  iunxpconst  4977  resiun2  5212  txbasval  19279  txtube  19313  txcmplem1  19314  ovoliunlem1  21085

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-sep 4495  ax-nul 4503  ax-pr 4613

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-ral 2797  df-rex 2798  df-v 3054  df-dif 3413  df-un 3415  df-in 3417  df-ss 3424  df-nul 3720  df-if 3874  df-sn 3960  df-pr 3962  df-op 3966  df-iun 4255  df-opab 4433  df-xp 4928