ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  xpssres Unicode version

Theorem xpssres 4734
Description: Restriction of a constant function (or other cross 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 4440 . . 3  |-  ( ( A  X.  B )  |`  C )  =  ( ( A  X.  B
)  i^i  ( C  X.  _V ) )
2 inxp 4558 . . 3  |-  ( ( A  X.  B )  i^i  ( C  X.  _V ) )  =  ( ( A  i^i  C
)  X.  ( B  i^i  _V ) )
3 incom 3190 . . . 4  |-  ( A  i^i  C )  =  ( C  i^i  A
)
4 inv1 3316 . . . 4  |-  ( B  i^i  _V )  =  B
53, 4xpeq12i 4450 . . 3  |-  ( ( A  i^i  C )  X.  ( B  i^i  _V ) )  =  ( ( C  i^i  A
)  X.  B )
61, 2, 53eqtri 2112 . 2  |-  ( ( A  X.  B )  |`  C )  =  ( ( C  i^i  A
)  X.  B )
7 df-ss 3010 . . . 4  |-  ( C 
C_  A  <->  ( C  i^i  A )  =  C )
87biimpi 118 . . 3  |-  ( C 
C_  A  ->  ( C  i^i  A )  =  C )
98xpeq1d 4451 . 2  |-  ( C 
C_  A  ->  (
( C  i^i  A
)  X.  B )  =  ( C  X.  B ) )
106, 9syl5eq 2132 1  |-  ( C 
C_  A  ->  (
( A  X.  B
)  |`  C )  =  ( C  X.  B
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1289   _Vcvv 2619    i^i cin 2996    C_ wss 2997    X. cxp 4426    |` cres 4430
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-opab 3892  df-xp 4434  df-rel 4435  df-res 4440
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator