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Theorem fconstmpt2 6334
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 4986 . 2  |-  ( ( A  X.  B )  X.  { C }
)  =  ( z  e.  ( A  X.  B )  |->  C )
2 eqidd 2403 . . 3  |-  ( z  =  <. x ,  y
>.  ->  C  =  C )
32mpt2mpt 6331 . 2  |-  ( z  e.  ( A  X.  B )  |->  C )  =  ( x  e.  A ,  y  e.  B  |->  C )
41, 3eqtri 2431 1  |-  ( ( A  X.  B )  X.  { C }
)  =  ( x  e.  A ,  y  e.  B  |->  C )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1405   {csn 3971   <.cop 3977    |-> cmpt 4452    X. cxp 4940    |-> cmpt2 6236
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-iun 4272  df-opab 4453  df-mpt 4454  df-xp 4948  df-rel 4949  df-oprab 6238  df-mpt2 6239
This theorem is referenced by:  tposconst  6950  mat0op  19105  matsc  19136  mdetrsca2  19290  smadiadetglem2  19358
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