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Theorem xpiindim 4646
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 4618 . . . . . 6  |-  Rel  ( C  X.  B )
21rgenw 2464 . . . . 5  |-  A. x  e.  A  Rel  ( C  X.  B )
3 r19.2m 3419 . . . . 5  |-  ( ( E. y  y  e.  A  /\  A. x  e.  A  Rel  ( C  X.  B ) )  ->  E. x  e.  A  Rel  ( C  X.  B
) )
42, 3mpan2 421 . . . 4  |-  ( E. y  y  e.  A  ->  E. x  e.  A  Rel  ( C  X.  B
) )
5 reliin 4631 . . . 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 4618 . . 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 2178 . . . . . . . 8  |-  ( x  =  y  ->  (
x  e.  A  <->  y  e.  A ) )
109cbvexv 1872 . . . . . . 7  |-  ( E. x  x  e.  A  <->  E. y  y  e.  A
)
11 r19.28mv 3425 . . . . . . 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 3788 . . . . . . 7  |-  ( z  e.  _V  ->  (
z  e.  |^|_ x  e.  A  B  <->  A. x  e.  A  z  e.  B ) )
1514elv 2664 . . . . . 6  |-  ( z  e.  |^|_ x  e.  A  B 
<-> 
A. x  e.  A  z  e.  B )
1615anbi2i 452 . . . . 5  |-  ( ( w  e.  C  /\  z  e.  |^|_ x  e.  A  B )  <->  ( w  e.  C  /\  A. x  e.  A  z  e.  B ) )
17 opelxp 4539 . . . . . 6  |-  ( <.
w ,  z >.  e.  ( C  X.  B
)  <->  ( w  e.  C  /\  z  e.  B ) )
1817ralbii 2418 . . . . 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 4539 . . . 4  |-  ( <.
w ,  z >.  e.  ( C  X.  |^|_ x  e.  A  B )  <-> 
( w  e.  C  /\  z  e.  |^|_ x  e.  A  B )
)
21 vex 2663 . . . . . 6  |-  w  e. 
_V
22 vex 2663 . . . . . 6  |-  z  e. 
_V
2321, 22opex 4121 . . . . 5  |-  <. w ,  z >.  e.  _V
24 eliin 3788 . . . . 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 4608 . 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 418 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 1316   E.wex 1453    e. wcel 1465   A.wral 2393   E.wrex 2394   _Vcvv 2660   <.cop 3500   |^|_ciin 3784    X. cxp 4507   Rel wrel 4514
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-iin 3786  df-opab 3960  df-xp 4515  df-rel 4516
This theorem is referenced by:  xpriindim  4647
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