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Theorem xpdisjres 23989
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 4849 . 2  |-  ( ( A  X.  B )  |`  C )  =  ( ( A  X.  B
)  i^i  ( C  X.  _V ) )
2 xpdisj1 5253 . 2  |-  ( ( A  i^i  C )  =  (/)  ->  ( ( A  X.  B )  i^i  ( C  X.  _V ) )  =  (/) )
31, 2syl5eq 2448 1  |-  ( ( A  i^i  C )  =  (/)  ->  ( ( A  X.  B )  |`  C )  =  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649   _Vcvv 2916    i^i cin 3279   (/)c0 3588    X. cxp 4835    |` cres 4839
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 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
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 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-opab 4227  df-xp 4843  df-rel 4844  df-res 4849
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