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Theorem xpssres 5156
Description: Restriction of a constant function (or other Cartesian product). (Contributed by Stefan O'Rear, 24-Jan-2015.)
Assertion
Ref Expression
xpssres  |-  ( C 
C_  A  ->  (
( A  X.  B
)  |`  C )  =  ( C  X.  B
) )

Proof of Theorem xpssres
StepHypRef Expression
1 df-res 4863 . . 3  |-  ( ( A  X.  B )  |`  C )  =  ( ( A  X.  B
)  i^i  ( C  X.  _V ) )
2 inxp 4984 . . 3  |-  ( ( A  X.  B )  i^i  ( C  X.  _V ) )  =  ( ( A  i^i  C
)  X.  ( B  i^i  _V ) )
3 incom 3656 . . . 4  |-  ( A  i^i  C )  =  ( C  i^i  A
)
4 inv1 3790 . . . 4  |-  ( B  i^i  _V )  =  B
53, 4xpeq12i 4873 . . 3  |-  ( ( A  i^i  C )  X.  ( B  i^i  _V ) )  =  ( ( C  i^i  A
)  X.  B )
61, 2, 53eqtri 2456 . 2  |-  ( ( A  X.  B )  |`  C )  =  ( ( C  i^i  A
)  X.  B )
7 df-ss 3451 . . . 4  |-  ( C 
C_  A  <->  ( C  i^i  A )  =  C )
87biimpi 198 . . 3  |-  ( C 
C_  A  ->  ( C  i^i  A )  =  C )
98xpeq1d 4874 . 2  |-  ( C 
C_  A  ->  (
( C  i^i  A
)  X.  B )  =  ( C  X.  B ) )
106, 9syl5eq 2476 1  |-  ( C 
C_  A  ->  (
( A  X.  B
)  |`  C )  =  ( C  X.  B
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1438   _Vcvv 3082    i^i cin 3436    C_ wss 3437    X. cxp 4849    |` cres 4853
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-sep 4544  ax-nul 4553  ax-pr 4658
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-rab 2785  df-v 3084  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3763  df-if 3911  df-sn 3998  df-pr 4000  df-op 4004  df-opab 4481  df-xp 4857  df-rel 4858  df-res 4863
This theorem is referenced by:  fparlem3  6907  fparlem4  6908  fpwwe2lem13  9069  pwssplit3  18277  cnconst2  20291  xkoccn  20626  tmdgsum  21102  dvcmul  22890  dvcmulf  22891  dvsconst  36543  dvsid  36544  aacllem  39846
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