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Theorem resmpo 5967
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 5966 . 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 5874 . . 3  |-  ( x  e.  A ,  y  e.  B  |->  E )  =  { <. <. x ,  y >. ,  z
>.  |  ( (
x  e.  A  /\  y  e.  B )  /\  z  =  E
) }
32reseq1i 4899 . 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 5874 . 2  |-  ( x  e.  C ,  y  e.  D  |->  E )  =  { <. <. x ,  y >. ,  z
>.  |  ( (
x  e.  C  /\  y  e.  D )  /\  z  =  E
) }
51, 3, 43eqtr4g 2235 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 104    = wceq 1353    e. wcel 2148    C_ wss 3129    X. cxp 4621    |` cres 4625   {coprab 5870    e. cmpo 5871
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-opab 4062  df-xp 4629  df-rel 4630  df-res 4635  df-oprab 5873  df-mpo 5874
This theorem is referenced by:  ofmres  6131  divfnzn  9607  txss12  13430  txbasval  13431  cnmpt2res  13461
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