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Theorem resmpt2 6375
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 6374 . 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 6280 . . 3  |-  ( x  e.  A ,  y  e.  B  |->  E )  =  { <. <. x ,  y >. ,  z
>.  |  ( (
x  e.  A  /\  y  e.  B )  /\  z  =  E
) }
32reseq1i 5260 . 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 6280 . 2  |-  ( x  e.  C ,  y  e.  D  |->  E )  =  { <. <. x ,  y >. ,  z
>.  |  ( (
x  e.  C  /\  y  e.  D )  /\  z  =  E
) }
51, 3, 43eqtr4g 2526 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1374    e. wcel 1762    C_ wss 3469    X. cxp 4990    |` cres 4994   {coprab 6276    |-> cmpt2 6277
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pr 4679
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-opab 4499  df-xp 4998  df-rel 4999  df-res 5004  df-oprab 6279  df-mpt2 6280
This theorem is referenced by:  ofmres  6770  cantnfval2  8077  cantnfval2OLD  8107  pgrpsubgsymg  16221  sylow3lem5  16440  mamures  18652  mdetrsca2  18866  mdetrlin2  18869  mdetunilem5  18878  smadiadetglem1  18933  smadiadetglem2  18934  pmatcollpw3lem  19044  txss12  19834  txbasval  19835  cnmpt2res  19906  fmucndlem  20522  cnmpt2pc  21156  oprpiece1res1  21179  oprpiece1res2  21180  cxpcn3  22843  ressplusf  27286  cvmlift2lem6  28379  cvmlift2lem12  28385
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