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Theorem xpiindim 4416
Description: Distributive law for cross product over indexed intersection. (Contributed by Jim Kingdon, 7-Dec-2018.)
Assertion
Ref Expression
xpiindim  C  X.  |^|_  |^|_  C  X.
Distinct variable groups:   ,,   , C,
Allowed substitution hints:   (,)

Proof of Theorem xpiindim
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relxp 4390 . . . . . 6  Rel  C  X.
21rgenw 2370 . . . . 5  Rel  C  X.
3 eleq1 2097 . . . . . . 7
43cbvexv 1792 . . . . . 6
5 r19.2m 3303 . . . . . 6  Rel  C  X.  Rel  C  X.
64, 5sylanbr 269 . . . . 5  Rel  C  X.  Rel  C  X.
72, 6mpan2 401 . . . 4  Rel  C  X.
8 reliin 4402 . . . 4  Rel  C  X.  Rel  |^|_  C  X.
97, 8syl 14 . . 3  Rel  |^|_  C  X.
10 relxp 4390 . . 3  Rel  C  X.  |^|_
119, 10jctil 295 . 2  Rel  C  X.  |^|_  Rel  |^|_  C  X.
12 r19.28mv 3308 . . . . . . 7  C  C
134, 12sylbir 125 . . . . . 6  C  C
1413bicomd 129 . . . . 5  C  C
15 vex 2554 . . . . . . 7 
_V
16 eliin 3653 . . . . . . 7  _V  |^|_
1715, 16ax-mp 7 . . . . . 6  |^|_
1817anbi2i 430 . . . . 5  C  |^|_  C
19 opelxp 4317 . . . . . 6  <. ,  >.  C  X.  C
2019ralbii 2324 . . . . 5  <. ,  >.  C  X.  C
2114, 18, 203bitr4g 212 . . . 4  C 
|^|_  <. , 
>.  C  X.
22 opelxp 4317 . . . 4  <. ,  >.  C  X.  |^|_  C  |^|_
23 vex 2554 . . . . . 6 
_V
2423, 15opex 3957 . . . . 5  <. ,  >.  _V
25 eliin 3653 . . . . 5  <. ,  >.  _V  <. , 
>.  |^|_  C  X.  <. ,  >.  C  X.
2624, 25ax-mp 7 . . . 4  <. ,  >.  |^|_  C  X.  <. , 
>.  C  X.
2721, 22, 263bitr4g 212 . . 3  <. , 
>.  C  X.  |^|_ 
<. ,  >. 
|^|_  C  X.
2827eqrelrdv2 4382 . 2  Rel  C  X.  |^|_  Rel  |^|_  C  X.  C  X.  |^|_  |^|_  C  X.
2911, 28mpancom 399 1  C  X.  |^|_  |^|_  C  X.
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wb 98   wceq 1242  wex 1378   wcel 1390  wral 2300  wrex 2301   _Vcvv 2551   <.cop 3370   |^|_ciin 3649    X. cxp 4286   Rel wrel 4293
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-iin 3651  df-opab 3810  df-xp 4294  df-rel 4295
This theorem is referenced by:  xpriindim  4417
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