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Theorem resmpt2 6109
Description: Restriction of the mapping operation. (Contributed by Mario Carneiro, 17-Dec-2013.)
Assertion
Ref Expression
resmpt2  |-  ( ( C  C_  A  /\  D  C_  B )  -> 
( ( x  e.  A ,  y  e.  B  |->  E )  |`  ( C  X.  D
) )  =  ( x  e.  C , 
y  e.  D  |->  E ) )
Distinct variable groups:    x, A, y    x, B, y    x, C, y    x, D, y
Allowed substitution hints:    E( x, y)

Proof of Theorem resmpt2
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 resoprab2 6108 . 2  |-  ( ( C  C_  A  /\  D  C_  B )  -> 
( { <. <. x ,  y >. ,  z
>.  |  ( (
x  e.  A  /\  y  e.  B )  /\  z  =  E
) }  |`  ( C  X.  D ) )  =  { <. <. x ,  y >. ,  z
>.  |  ( (
x  e.  C  /\  y  e.  D )  /\  z  =  E
) } )
2 df-mpt2 6027 . . 3  |-  ( x  e.  A ,  y  e.  B  |->  E )  =  { <. <. x ,  y >. ,  z
>.  |  ( (
x  e.  A  /\  y  e.  B )  /\  z  =  E
) }
32reseq1i 5084 . 2  |-  ( ( x  e.  A , 
y  e.  B  |->  E )  |`  ( C  X.  D ) )  =  ( { <. <. x ,  y >. ,  z
>.  |  ( (
x  e.  A  /\  y  e.  B )  /\  z  =  E
) }  |`  ( C  X.  D ) )
4 df-mpt2 6027 . 2  |-  ( x  e.  C ,  y  e.  D  |->  E )  =  { <. <. x ,  y >. ,  z
>.  |  ( (
x  e.  C  /\  y  e.  D )  /\  z  =  E
) }
51, 3, 43eqtr4g 2446 1  |-  ( ( C  C_  A  /\  D  C_  B )  -> 
( ( x  e.  A ,  y  e.  B  |->  E )  |`  ( C  X.  D
) )  =  ( x  e.  C , 
y  e.  D  |->  E ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717    C_ wss 3265    X. cxp 4818    |` cres 4822   {coprab 6023    e. cmpt2 6024
This theorem is referenced by:  ofmres  6284  cantnfval2  7559  sylow3lem5  15194  txss12  17560  txbasval  17561  cnmpt2res  17632  fmucndlem  18244  cnmpt2pc  18826  oprpiece1res1  18849  oprpiece1res2  18850  cxpcn3  20501  ressplusf  24024  cvmlift2lem6  24776  cvmlift2lem12  24782
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  df-oprab 6026  df-mpt2 6027
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