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Theorem xpiindi 4989
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 4961 . . . . . 6  |-  Rel  ( C  X.  B )
21rgenw 2783 . . . . 5  |-  A. x  e.  A  Rel  ( C  X.  B )
3 r19.2z 3888 . . . . 5  |-  ( ( A  =/=  (/)  /\  A. x  e.  A  Rel  ( C  X.  B
) )  ->  E. x  e.  A  Rel  ( C  X.  B ) )
42, 3mpan2 675 . . . 4  |-  ( A  =/=  (/)  ->  E. x  e.  A  Rel  ( C  X.  B ) )
5 reliin 4974 . . . 4  |-  ( E. x  e.  A  Rel  ( C  X.  B
)  ->  Rel  |^|_ x  e.  A  ( C  X.  B ) )
64, 5syl 17 . . 3  |-  ( A  =/=  (/)  ->  Rel  |^|_ x  e.  A  ( C  X.  B ) )
7 relxp 4961 . . 3  |-  Rel  ( C  X.  |^|_ x  e.  A  B )
86, 7jctil 539 . 2  |-  ( A  =/=  (/)  ->  ( Rel  ( C  X.  |^|_ x  e.  A  B )  /\  Rel  |^|_ x  e.  A  ( C  X.  B
) ) )
9 r19.28zv 3894 . . . . . 6  |-  ( A  =/=  (/)  ->  ( A. x  e.  A  (
y  e.  C  /\  z  e.  B )  <->  ( y  e.  C  /\  A. x  e.  A  z  e.  B ) ) )
109bicomd 204 . . . . 5  |-  ( A  =/=  (/)  ->  ( (
y  e.  C  /\  A. x  e.  A  z  e.  B )  <->  A. x  e.  A  ( y  e.  C  /\  z  e.  B ) ) )
11 vex 3083 . . . . . . 7  |-  z  e. 
_V
12 eliin 4305 . . . . . . 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 698 . . . . 5  |-  ( ( y  e.  C  /\  z  e.  |^|_ x  e.  A  B )  <->  ( y  e.  C  /\  A. x  e.  A  z  e.  B ) )
15 opelxp 4883 . . . . . 6  |-  ( <.
y ,  z >.  e.  ( C  X.  B
)  <->  ( y  e.  C  /\  z  e.  B ) )
1615ralbii 2853 . . . . 5  |-  ( A. x  e.  A  <. y ,  z >.  e.  ( C  X.  B )  <->  A. x  e.  A  ( y  e.  C  /\  z  e.  B
) )
1710, 14, 163bitr4g 291 . . . 4  |-  ( A  =/=  (/)  ->  ( (
y  e.  C  /\  z  e.  |^|_ x  e.  A  B )  <->  A. x  e.  A  <. y ,  z >.  e.  ( C  X.  B ) ) )
18 opelxp 4883 . . . 4  |-  ( <.
y ,  z >.  e.  ( C  X.  |^|_ x  e.  A  B )  <-> 
( y  e.  C  /\  z  e.  |^|_ x  e.  A  B )
)
19 opex 4685 . . . . 5  |-  <. y ,  z >.  e.  _V
20 eliin 4305 . . . . 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 291 . . 3  |-  ( A  =/=  (/)  ->  ( <. y ,  z >.  e.  ( C  X.  |^|_ x  e.  A  B )  <->  <.
y ,  z >.  e.  |^|_ x  e.  A  ( C  X.  B
) ) )
2322eqrelrdv2 4953 . 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 673 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 187    /\ wa 370    = wceq 1437    e. wcel 1872    =/= wne 2614   A.wral 2771   E.wrex 2772   _Vcvv 3080   (/)c0 3761   <.cop 4004   |^|_ciin 4300    X. cxp 4851   Rel wrel 4858
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pr 4660
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-sn 3999  df-pr 4001  df-op 4005  df-iin 4302  df-opab 4483  df-xp 4859  df-rel 4860
This theorem is referenced by:  xpriindi  4990
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