Theorem xpiundir 5060

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 3129	. . . . 5
2		df-rex 2813	. . . . . 6
3	2	rexbii 2959	. . . . 5
4		eliun 4335	. . . . . . . 8
5	4	anbi1i 695	. . . . . . 7
6		r19.41v 3009	. . . . . . 7
7	5, 6	bitr4i 252	. . . . . 6
8	7	exbii 1667	. . . . 5
9	1, 3, 8	3bitr4ri 278	. . . 4
10		df-rex 2813	. . . 4
11		elxp2 5022	. . . . 5
12	11	rexbii 2959	. . . 4
13	9, 10, 12	3bitr4i 277	. . 3
14		elxp2 5022	. . 3
15		eliun 4335	. . 3
16	13, 14, 15	3bitr4i 277	. 2
17	16	eqriv 2453	1

Syntax hints:  /\wa 369  =wceq 1395  E.wex 1612  e.wcel 1818  E.wrex 2808  <.cop 4035  U_ciun 4330  X.cxp 5002

This theorem is referenced by:  iunxpconst  5061  resiun2  5298  txbasval  20107  txtube  20141  txcmplem1  20142  ovoliunlem1  21913

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-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-iun 4332  df-opab 4511  df-xp 5010