Theorem xpiundir 4819

Description: Distributive law for cross product over indexed union. (Contributed by set.mm contributors, 26-Apr-2014.) (Revised by Mario Carneiro, 27-Apr-2014.)

Assertion

Ref	Expression
xpiundir	|- (U_x e. A B X. C) = U_x e. A (B X. C)

Distinct variable group:   x,C

Allowed substitution hints:   A(x)   B(x)

Proof of Theorem xpiundir

Dummy variables y w z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rexcom4 2879	. . . . 5 |- (E.x e. A E.y(y e. B /\ E.w e. C z = <.y, w>.) <-> E.yE.x e. A (y e. B /\ E.w e. C z = <.y, w>.))
2		df-rex 2621	. . . . . 6 |- (E.y e. B E.w e. C z = <.y, w>. <-> E.y(y e. B /\ E.w e. C z = <.y, w>.))
3	2	rexbii 2640	. . . . 5 |- (E.x e. A E.y e. B E.w e. C z = <.y, w>. <-> E.x e. A E.y(y e. B /\ E.w e. C z = <.y, w>.))
4		eliun 3974	. . . . . . . 8 |- (y e. U_x e. A B <-> E.x e. A y e. B)
5	4	anbi1i 676	. . . . . . 7 |- ((y e. U_x e. A B /\ E.w e. C z = <.y, w>.) <-> (E.x e. A y e. B /\ E.w e. C z = <.y, w>.))
6		r19.41v 2765	. . . . . . 7 |- (E.x e. A (y e. B /\ E.w e. C z = <.y, w>.) <-> (E.x e. A y e. B /\ E.w e. C z = <.y, w>.))
7	5, 6	bitr4i 243	. . . . . 6 |- ((y e. U_x e. A B /\ E.w e. C z = <.y, w>.) <-> E.x e. A (y e. B /\ E.w e. C z = <.y, w>.))
8	7	exbii 1582	. . . . 5 |- (E.y(y e. U_x e. A B /\ E.w e. C z = <.y, w>.) <-> E.yE.x e. A (y e. B /\ E.w e. C z = <.y, w>.))
9	1, 3, 8	3bitr4ri 269	. . . 4 |- (E.y(y e. U_x e. A B /\ E.w e. C z = <.y, w>.) <-> E.x e. A E.y e. B E.w e. C z = <.y, w>.)
10		df-rex 2621	. . . 4 |- (E.y e. U_ x e. A BE.w e. C z = <.y, w>. <-> E.y(y e. U_x e. A B /\ E.w e. C z = <.y, w>.))
11		elxp2 4803	. . . . 5 |- (z e. (B X. C) <-> E.y e. B E.w e. C z = <.y, w>.)
12	11	rexbii 2640	. . . 4 |- (E.x e. A z e. (B X. C) <-> E.x e. A E.y e. B E.w e. C z = <.y, w>.)
13	9, 10, 12	3bitr4i 268	. . 3 |- (E.y e. U_ x e. A BE.w e. C z = <.y, w>. <-> E.x e. A z e. (B X. C))
14		elxp2 4803	. . 3 |- (z e. (U_x e. A B X. C) <-> E.y e. U_ x e. A BE.w e. C z = <.y, w>.)
15		eliun 3974	. . 3 |- (z e. U_x e. A (B X. C) <-> E.x e. A z e. (B X. C))
16	13, 14, 15	3bitr4i 268	. 2 |- (z e. (U_x e. A B X. C) <-> z e. U_x e. A (B X. C))
17	16	eqriv 2350	1 |- (U_x e. A B X. C) = U_x e. A (B X. C)

Syntax hints:   /\ wa 358  E.wex 1541   = wceq 1642   e. wcel 1710  E.wrex 2616  U_ciun 3970  <.cop 4562   X. cxp 4771

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-iun 3972  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-addc 4379  df-nnc 4380  df-phi 4566  df-op 4567  df-opab 4624  df-xp 4785

This theorem is referenced by:  iunxpconst  4820