Theorem xpiindi 5257

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

Assertion

Ref	Expression
xpiindi	|- ( A =/= (/) -> ( C X. |^|_ x e. A B ) = |^|_ x e. A ( C X. B ) )

Distinct variable groups:   x, A   x, C

Allowed substitution hint:   B( x)

Proof of Theorem xpiindi

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

Step	Hyp	Ref	Expression
1		relxp 5227	. . . . . 6 |- Rel ( C X. B )
2	1	rgenw 2924	. . . . 5 |- A. x e. A Rel ( C X. B )
3		r19.2z 4060	. . . . 5 |- ( ( A =/= (/) /\ A. x e. A Rel ( C X. B ) ) -> E. x e. A Rel ( C X. B ) )
4	2, 3	mpan2 707	. . . 4 |- ( A =/= (/) -> E. x e. A Rel ( C X. B ) )
5		reliin 5240	. . . 4 |- ( E. x e. A Rel ( C X. B ) -> Rel |^|_ x e. A ( C X. B ) )
6	4, 5	syl 17	. . 3 |- ( A =/= (/) -> Rel |^|_ x e. A ( C X. B ) )
7		relxp 5227	. . 3 |- Rel ( C X. |^|_ x e. A B )
8	6, 7	jctil 560	. 2 |- ( A =/= (/) -> ( Rel ( C X. |^|_ x e. A B ) /\ Rel |^|_ x e. A ( C X. B ) ) )
9		r19.28zv 4066	. . . . . 6 |- ( A =/= (/) -> ( A. x e. A ( y e. C /\ z e. B ) <-> ( y e. C /\ A. x e. A z e. B ) ) )
10	9	bicomd 213	. . . . 5 |- ( A =/= (/) -> ( ( y e. C /\ A. x e. A z e. B ) <-> A. x e. A ( y e. C /\ z e. B ) ) )
11		vex 3203	. . . . . . 7 |- z e. _V
12		eliin 4525	. . . . . . 7 |- ( z e. _V -> ( z e. |^|_ x e. A B <-> A. x e. A z e. B ) )
13	11, 12	ax-mp 5	. . . . . 6 |- ( z e. |^|_ x e. A B <-> A. x e. A z e. B )
14	13	anbi2i 730	. . . . 5 |- ( ( y e. C /\ z e. |^|_ x e. A B ) <-> ( y e. C /\ A. x e. A z e. B ) )
15		opelxp 5146	. . . . . 6 |- ( <. y , z >. e. ( C X. B ) <-> ( y e. C /\ z e. B ) )
16	15	ralbii 2980	. . . . 5 |- ( A. x e. A <. y , z >. e. ( C X. B ) <-> A. x e. A ( y e. C /\ z e. B ) )
17	10, 14, 16	3bitr4g 303	. . . 4 |- ( A =/= (/) -> ( ( y e. C /\ z e. |^|_ x e. A B ) <-> A. x e. A <. y , z >. e. ( C X. B ) ) )
18		opelxp 5146	. . . 4 |- ( <. y , z >. e. ( C X. |^|_ x e. A B ) <-> ( y e. C /\ z e. |^|_ x e. A B ) )
19		opex 4932	. . . . 5 |- <. y , z >. e. _V
20		eliin 4525	. . . . 5 |- ( <. y , z >. e. _V -> ( <. y , z >. e. |^|_ x e. A ( C X. B ) <-> A. x e. A <. y , z >. e. ( C X. B ) ) )
21	19, 20	ax-mp 5	. . . 4 |- ( <. y , z >. e. |^|_ x e. A ( C X. B ) <-> A. x e. A <. y , z >. e. ( C X. B ) )
22	17, 18, 21	3bitr4g 303	. . 3 |- ( A =/= (/) -> ( <. y , z >. e. ( C X. |^|_ x e. A B ) <-> <. y , z >. e. |^|_ x e. A ( C X. B ) ) )
23	22	eqrelrdv2 5219	. 2 |- ( ( ( Rel ( C X. |^|_ x e. A B ) /\ Rel |^|_ x e. A ( C X. B ) ) /\ A =/= (/) ) -> ( C X. |^|_ x e. A B ) = |^|_ x e. A ( C X. B ) )
24	8, 23	mpancom 703	1 |- ( A =/= (/) -> ( C X. |^|_ x e. A B ) = |^|_ x e. A ( C X. B ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   _Vcvv 3200   (/)c0 3915   <.cop 4183   |^|_ciin 4521   X. cxp 5112   Rel wrel 5119

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-iin 4523  df-opab 4713  df-xp 5120  df-rel 5121

This theorem is referenced by:  xpriindi  5258