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Theorem resmpo 5835
Description: Restriction of the mapping operation. (Contributed by Mario Carneiro, 17-Dec-2013.)
Assertion
Ref Expression
resmpo  |-  ( ( 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 resmpo
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 resoprab2 5834 . 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-mpo 5745 . . 3  |-  ( x  e.  A ,  y  e.  B  |->  E )  =  { <. <. x ,  y >. ,  z
>.  |  ( (
x  e.  A  /\  y  e.  B )  /\  z  =  E
) }
32reseq1i 4783 . 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-mpo 5745 . 2  |-  ( x  e.  C ,  y  e.  D  |->  E )  =  { <. <. x ,  y >. ,  z
>.  |  ( (
x  e.  C  /\  y  e.  D )  /\  z  =  E
) }
51, 3, 43eqtr4g 2173 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 103    = wceq 1314    e. wcel 1463    C_ wss 3039    X. cxp 4505    |` cres 4509   {coprab 5741    e. cmpo 5742
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-opab 3958  df-xp 4513  df-rel 4514  df-res 4519  df-oprab 5744  df-mpo 5745
This theorem is referenced by:  ofmres  6000  divfnzn  9365  txss12  12341  txbasval  12342  cnmpt2res  12372
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