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Theorem xpdisjres 23881
Description: Restriction of a constant function (or other cross product) outside of its domain (Contributed by Thierry Arnoux, 25-Jan-2017.)
Assertion
Ref Expression
xpdisjres  |-  ( ( A  i^i  C )  =  (/)  ->  ( ( A  X.  B )  |`  C )  =  (/) )

Proof of Theorem xpdisjres
StepHypRef Expression
1 df-res 4832 . 2  |-  ( ( A  X.  B )  |`  C )  =  ( ( A  X.  B
)  i^i  ( C  X.  _V ) )
2 xpdisj1 5236 . 2  |-  ( ( A  i^i  C )  =  (/)  ->  ( ( A  X.  B )  i^i  ( C  X.  _V ) )  =  (/) )
31, 2syl5eq 2433 1  |-  ( ( A  i^i  C )  =  (/)  ->  ( ( A  X.  B )  |`  C )  =  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649   _Vcvv 2901    i^i cin 3264   (/)c0 3573    X. cxp 4818    |` cres 4822
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pr 4346
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-op 3768  df-opab 4210  df-xp 4826  df-rel 4827  df-res 4832
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