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Theorem xpdisjres 27669
Description: Restriction of a constant function (or other Cartesian 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 5000 . 2  |-  ( ( A  X.  B )  |`  C )  =  ( ( A  X.  B
)  i^i  ( C  X.  _V ) )
2 xpdisj1 5413 . 2  |-  ( ( A  i^i  C )  =  (/)  ->  ( ( A  X.  B )  i^i  ( C  X.  _V ) )  =  (/) )
31, 2syl5eq 2507 1  |-  ( ( A  i^i  C )  =  (/)  ->  ( ( A  X.  B )  |`  C )  =  (/) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1398   _Vcvv 3106    i^i cin 3460   (/)c0 3783    X. cxp 4986    |` cres 4990
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-opab 4498  df-xp 4994  df-rel 4995  df-res 5000
This theorem is referenced by:  padct  27776
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