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Theorem resmpt 4957
Description: Restriction of the mapping operation. (Contributed by Mario Carneiro, 15-Jul-2013.)
Assertion
Ref Expression
resmpt  |-  ( B 
C_  A  ->  (
( x  e.  A  |->  C )  |`  B )  =  ( x  e.  B  |->  C ) )
Distinct variable groups:    x, A    x, B
Allowed substitution hint:    C( x)

Proof of Theorem resmpt
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 resopab2 4956 . 2  |-  ( B 
C_  A  ->  ( { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  C ) }  |`  B )  =  { <. x ,  y >.  |  ( x  e.  B  /\  y  =  C ) } )
2 df-mpt 4068 . . 3  |-  ( x  e.  A  |->  C )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  C ) }
32reseq1i 4905 . 2  |-  ( ( x  e.  A  |->  C )  |`  B )  =  ( { <. x ,  y >.  |  ( x  e.  A  /\  y  =  C ) }  |`  B )
4 df-mpt 4068 . 2  |-  ( x  e.  B  |->  C )  =  { <. x ,  y >.  |  ( x  e.  B  /\  y  =  C ) }
51, 3, 43eqtr4g 2235 1  |-  ( B 
C_  A  ->  (
( x  e.  A  |->  C )  |`  B )  =  ( x  e.  B  |->  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1353    e. wcel 2148    C_ wss 3131   {copab 4065    |-> cmpt 4066    |` cres 4630
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 4123  ax-pow 4176  ax-pr 4211
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 2741  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-opab 4067  df-mpt 4068  df-xp 4634  df-rel 4635  df-res 4640
This theorem is referenced by:  resmpt3  4958  resmptf  4959  resmptd  4960  f1stres  6162  f2ndres  6163  tposss  6249  dftpos2  6264  dftpos4  6266  djuf1olemr  7055  fisumss  11402  isumclim3  11433  expcnv  11514  fprodssdc  11600  tgrest  13708  cnmptid  13820  dvidlemap  14199  dvcnp2cntop  14202  dvmulxxbr  14205  dvcoapbr  14210  dvrecap  14216
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