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Theorem fconstmpt2 6374
Description: Representation of a constant operation using the mapping operation. (Contributed by SO, 11-Jul-2018.)
Assertion
Ref Expression
fconstmpt2  |-  ( ( A  X.  B )  X.  { C }
)  =  ( x  e.  A ,  y  e.  B  |->  C )
Distinct variable groups:    x, A, y    x, B, y    x, C, y

Proof of Theorem fconstmpt2
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 fconstmpt 5037 . 2  |-  ( ( A  X.  B )  X.  { C }
)  =  ( z  e.  ( A  X.  B )  |->  C )
2 eqidd 2463 . . 3  |-  ( z  =  <. x ,  y
>.  ->  C  =  C )
32mpt2mpt 6371 . 2  |-  ( z  e.  ( A  X.  B )  |->  C )  =  ( x  e.  A ,  y  e.  B  |->  C )
41, 3eqtri 2491 1  |-  ( ( A  X.  B )  X.  { C }
)  =  ( x  e.  A ,  y  e.  B  |->  C )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1374   {csn 4022   <.cop 4028    |-> cmpt 4500    X. cxp 4992    |-> cmpt2 6279
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 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pr 4681
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 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-sn 4023  df-pr 4025  df-op 4029  df-iun 4322  df-opab 4501  df-mpt 4502  df-xp 5000  df-rel 5001  df-oprab 6281  df-mpt2 6282
This theorem is referenced by:  tposconst  6985  mat0op  18683  matsc  18714  mdetrsca2  18868  smadiadetglem2  18936
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