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Theorem xpiindim 4723
Description: Distributive law for cross product over indexed intersection. (Contributed by Jim Kingdon, 7-Dec-2018.)
Assertion
Ref Expression
xpiindim  |-  ( E. y  y  e.  A  ->  ( C  X.  |^|_ x  e.  A  B )  =  |^|_ x  e.  A  ( C  X.  B
) )
Distinct variable groups:    x, y, A   
x, C, y
Allowed substitution hints:    B( x, y)

Proof of Theorem xpiindim
Dummy variables  z  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relxp 4695 . . . . . 6  |-  Rel  ( C  X.  B )
21rgenw 2512 . . . . 5  |-  A. x  e.  A  Rel  ( C  X.  B )
3 r19.2m 3480 . . . . 5  |-  ( ( E. y  y  e.  A  /\  A. x  e.  A  Rel  ( C  X.  B ) )  ->  E. x  e.  A  Rel  ( C  X.  B
) )
42, 3mpan2 422 . . . 4  |-  ( E. y  y  e.  A  ->  E. x  e.  A  Rel  ( C  X.  B
) )
5 reliin 4708 . . . 4  |-  ( E. x  e.  A  Rel  ( C  X.  B
)  ->  Rel  |^|_ x  e.  A  ( C  X.  B ) )
64, 5syl 14 . . 3  |-  ( E. y  y  e.  A  ->  Rel  |^|_ x  e.  A  ( C  X.  B
) )
7 relxp 4695 . . 3  |-  Rel  ( C  X.  |^|_ x  e.  A  B )
86, 7jctil 310 . 2  |-  ( E. y  y  e.  A  ->  ( Rel  ( C  X.  |^|_ x  e.  A  B )  /\  Rel  |^|_
x  e.  A  ( C  X.  B ) ) )
9 eleq1w 2218 . . . . . . . 8  |-  ( x  =  y  ->  (
x  e.  A  <->  y  e.  A ) )
109cbvexv 1898 . . . . . . 7  |-  ( E. x  x  e.  A  <->  E. y  y  e.  A
)
11 r19.28mv 3486 . . . . . . 7  |-  ( E. x  x  e.  A  ->  ( A. x  e.  A  ( w  e.  C  /\  z  e.  B )  <->  ( w  e.  C  /\  A. x  e.  A  z  e.  B ) ) )
1210, 11sylbir 134 . . . . . 6  |-  ( E. y  y  e.  A  ->  ( A. x  e.  A  ( w  e.  C  /\  z  e.  B )  <->  ( w  e.  C  /\  A. x  e.  A  z  e.  B ) ) )
1312bicomd 140 . . . . 5  |-  ( E. y  y  e.  A  ->  ( ( w  e.  C  /\  A. x  e.  A  z  e.  B )  <->  A. x  e.  A  ( w  e.  C  /\  z  e.  B ) ) )
14 eliin 3854 . . . . . . 7  |-  ( z  e.  _V  ->  (
z  e.  |^|_ x  e.  A  B  <->  A. x  e.  A  z  e.  B ) )
1514elv 2716 . . . . . 6  |-  ( z  e.  |^|_ x  e.  A  B 
<-> 
A. x  e.  A  z  e.  B )
1615anbi2i 453 . . . . 5  |-  ( ( w  e.  C  /\  z  e.  |^|_ x  e.  A  B )  <->  ( w  e.  C  /\  A. x  e.  A  z  e.  B ) )
17 opelxp 4616 . . . . . 6  |-  ( <.
w ,  z >.  e.  ( C  X.  B
)  <->  ( w  e.  C  /\  z  e.  B ) )
1817ralbii 2463 . . . . 5  |-  ( A. x  e.  A  <. w ,  z >.  e.  ( C  X.  B )  <->  A. x  e.  A  ( w  e.  C  /\  z  e.  B
) )
1913, 16, 183bitr4g 222 . . . 4  |-  ( E. y  y  e.  A  ->  ( ( w  e.  C  /\  z  e. 
|^|_ x  e.  A  B )  <->  A. x  e.  A  <. w ,  z >.  e.  ( C  X.  B ) ) )
20 opelxp 4616 . . . 4  |-  ( <.
w ,  z >.  e.  ( C  X.  |^|_ x  e.  A  B )  <-> 
( w  e.  C  /\  z  e.  |^|_ x  e.  A  B )
)
21 vex 2715 . . . . . 6  |-  w  e. 
_V
22 vex 2715 . . . . . 6  |-  z  e. 
_V
2321, 22opex 4189 . . . . 5  |-  <. w ,  z >.  e.  _V
24 eliin 3854 . . . . 5  |-  ( <.
w ,  z >.  e.  _V  ->  ( <. w ,  z >.  e.  |^|_ x  e.  A  ( C  X.  B )  <->  A. x  e.  A  <. w ,  z >.  e.  ( C  X.  B ) ) )
2523, 24ax-mp 5 . . . 4  |-  ( <.
w ,  z >.  e.  |^|_ x  e.  A  ( C  X.  B
)  <->  A. x  e.  A  <. w ,  z >.  e.  ( C  X.  B
) )
2619, 20, 253bitr4g 222 . . 3  |-  ( E. y  y  e.  A  ->  ( <. w ,  z
>.  e.  ( C  X.  |^|_
x  e.  A  B
)  <->  <. w ,  z
>.  e.  |^|_ x  e.  A  ( C  X.  B
) ) )
2726eqrelrdv2 4685 . 2  |-  ( ( ( Rel  ( C  X.  |^|_ x  e.  A  B )  /\  Rel  |^|_
x  e.  A  ( C  X.  B ) )  /\  E. y 
y  e.  A )  ->  ( C  X.  |^|_
x  e.  A  B
)  =  |^|_ x  e.  A  ( C  X.  B ) )
288, 27mpancom 419 1  |-  ( E. y  y  e.  A  ->  ( C  X.  |^|_ x  e.  A  B )  =  |^|_ x  e.  A  ( C  X.  B
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1335   E.wex 1472    e. wcel 2128   A.wral 2435   E.wrex 2436   _Vcvv 2712   <.cop 3563   |^|_ciin 3850    X. cxp 4584   Rel wrel 4591
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-iin 3852  df-opab 4026  df-xp 4592  df-rel 4593
This theorem is referenced by:  xpriindim  4724
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