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Theorem xpiindi 5076
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.
StepHypRef Expression
1 relxp 5048 . . . . . 6  |-  Rel  ( C  X.  B )
21rgenw 2894 . . . . 5  |-  A. x  e.  A  Rel  ( C  X.  B )
3 r19.2z 3870 . . . . 5  |-  ( ( A  =/=  (/)  /\  A. x  e.  A  Rel  ( C  X.  B
) )  ->  E. x  e.  A  Rel  ( C  X.  B ) )
42, 3mpan2 671 . . . 4  |-  ( A  =/=  (/)  ->  E. x  e.  A  Rel  ( C  X.  B ) )
5 reliin 5062 . . . 4  |-  ( E. x  e.  A  Rel  ( C  X.  B
)  ->  Rel  |^|_ x  e.  A  ( C  X.  B ) )
64, 5syl 16 . . 3  |-  ( A  =/=  (/)  ->  Rel  |^|_ x  e.  A  ( C  X.  B ) )
7 relxp 5048 . . 3  |-  Rel  ( C  X.  |^|_ x  e.  A  B )
86, 7jctil 537 . 2  |-  ( A  =/=  (/)  ->  ( Rel  ( C  X.  |^|_ x  e.  A  B )  /\  Rel  |^|_ x  e.  A  ( C  X.  B
) ) )
9 r19.28zv 3876 . . . . . 6  |-  ( A  =/=  (/)  ->  ( A. x  e.  A  (
y  e.  C  /\  z  e.  B )  <->  ( y  e.  C  /\  A. x  e.  A  z  e.  B ) ) )
109bicomd 201 . . . . 5  |-  ( A  =/=  (/)  ->  ( (
y  e.  C  /\  A. x  e.  A  z  e.  B )  <->  A. x  e.  A  ( y  e.  C  /\  z  e.  B ) ) )
11 vex 3074 . . . . . . 7  |-  z  e. 
_V
12 eliin 4277 . . . . . . 7  |-  ( z  e.  _V  ->  (
z  e.  |^|_ x  e.  A  B  <->  A. x  e.  A  z  e.  B ) )
1311, 12ax-mp 5 . . . . . 6  |-  ( z  e.  |^|_ x  e.  A  B 
<-> 
A. x  e.  A  z  e.  B )
1413anbi2i 694 . . . . 5  |-  ( ( y  e.  C  /\  z  e.  |^|_ x  e.  A  B )  <->  ( y  e.  C  /\  A. x  e.  A  z  e.  B ) )
15 opelxp 4970 . . . . . 6  |-  ( <.
y ,  z >.  e.  ( C  X.  B
)  <->  ( y  e.  C  /\  z  e.  B ) )
1615ralbii 2834 . . . . 5  |-  ( A. x  e.  A  <. y ,  z >.  e.  ( C  X.  B )  <->  A. x  e.  A  ( y  e.  C  /\  z  e.  B
) )
1710, 14, 163bitr4g 288 . . . 4  |-  ( A  =/=  (/)  ->  ( (
y  e.  C  /\  z  e.  |^|_ x  e.  A  B )  <->  A. x  e.  A  <. y ,  z >.  e.  ( C  X.  B ) ) )
18 opelxp 4970 . . . 4  |-  ( <.
y ,  z >.  e.  ( C  X.  |^|_ x  e.  A  B )  <-> 
( y  e.  C  /\  z  e.  |^|_ x  e.  A  B )
)
19 opex 4657 . . . . 5  |-  <. y ,  z >.  e.  _V
20 eliin 4277 . . . . 5  |-  ( <.
y ,  z >.  e.  _V  ->  ( <. y ,  z >.  e.  |^|_ x  e.  A  ( C  X.  B )  <->  A. x  e.  A  <. y ,  z >.  e.  ( C  X.  B ) ) )
2119, 20ax-mp 5 . . . 4  |-  ( <.
y ,  z >.  e.  |^|_ x  e.  A  ( C  X.  B
)  <->  A. x  e.  A  <. y ,  z >.  e.  ( C  X.  B
) )
2217, 18, 213bitr4g 288 . . 3  |-  ( A  =/=  (/)  ->  ( <. y ,  z >.  e.  ( C  X.  |^|_ x  e.  A  B )  <->  <.
y ,  z >.  e.  |^|_ x  e.  A  ( C  X.  B
) ) )
2322eqrelrdv2 5040 . 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 ) )
248, 23mpancom 669 1  |-  ( A  =/=  (/)  ->  ( C  X.  |^|_ x  e.  A  B )  =  |^|_ x  e.  A  ( C  X.  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2644   A.wral 2795   E.wrex 2796   _Vcvv 3071   (/)c0 3738   <.cop 3984   |^|_ciin 4273    X. cxp 4939   Rel wrel 4946
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 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pr 4632
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 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-sn 3979  df-pr 3981  df-op 3985  df-iin 4275  df-opab 4452  df-xp 4947  df-rel 4948
This theorem is referenced by:  xpriindi  5077
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