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Theorem fconstmpt 4415
Description: Representation of a constant function using the mapping operation. (Note that  x cannot appear free in  B.) (Contributed by NM, 12-Oct-1999.) (Revised by Mario Carneiro, 16-Nov-2013.)
Assertion
Ref Expression
fconstmpt  |-  ( A  X.  { B }
)  =  ( x  e.  A  |->  B )
Distinct variable groups:    x, A    x, B

Proof of Theorem fconstmpt
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 velsn 3420 . . . 4  |-  ( y  e.  { B }  <->  y  =  B )
21anbi2i 438 . . 3  |-  ( ( x  e.  A  /\  y  e.  { B } )  <->  ( x  e.  A  /\  y  =  B ) )
32opabbii 3852 . 2  |-  { <. x ,  y >.  |  ( x  e.  A  /\  y  e.  { B } ) }  =  { <. x ,  y
>.  |  ( x  e.  A  /\  y  =  B ) }
4 df-xp 4379 . 2  |-  ( A  X.  { B }
)  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  e.  { B } ) }
5 df-mpt 3848 . 2  |-  ( x  e.  A  |->  B )  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  =  B ) }
63, 4, 53eqtr4i 2086 1  |-  ( A  X.  { B }
)  =  ( x  e.  A  |->  B )
Colors of variables: wff set class
Syntax hints:    /\ wa 101    = wceq 1259    e. wcel 1409   {csn 3403   {copab 3845    |-> cmpt 3846    X. cxp 4371
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-sn 3409  df-opab 3847  df-mpt 3848  df-xp 4379
This theorem is referenced by:  fconst  5110  fcoconst  5362  fmptsn  5380  ofc12  5759  caofinvl  5761  xpexgALT  5788
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