Theorem fconstmpt 4824

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.

Step	Hyp	Ref	Expression
1		velsn 3726	. . . 4 |- ( y e. { B } <-> y = B )
2	1	anbi2i 461	. . 3 |- ( ( x e. A /\ y e. { B } ) <-> ( x e. A /\ y = B ) )
3	2	opabbii 4199	. 2 |- { <. x , y >. | ( x e. A /\ y e. { B } ) } = { <. x , y >. | ( x e. A /\ y = B ) }
4		df-xp 4782	. 2 |- ( A X. { B } ) = { <. x , y >. | ( x e. A /\ y e. { B } ) }
5		df-mpt 4195	. 2 |- ( x e. A |-> B ) = { <. x , y >. | ( x e. A /\ y = B ) }
6	3, 4, 5	3eqtr4i 2269	1 |- ( A X. { B } ) = ( x e. A |-> B )

This proof depends on syntax axioms:   /\ wa 104   = wceq 1402   e. wcel 2209   {csn 3709   {copab 4192   |-> cmpt 4193   X. cxp 4774

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-sn 3715  df-opab 4194  df-mpt 4195  df-xp 4782

This theorem is used by:  fconst  5591  fcoconst  5882  fmptsn  5907  fconstmpo  6186  ofc12  6330  caofinvl  6332  xpexgALT  6370  inftonninf  10899  fser0const  10992  prod1dc  12377  gsumconstcmn  14265  pws0g  14319  rrgsupp  14682  psrlinv  15203  psr1clfi  15207  psrlidmfi  15208  psrridmfi  15209  mpl0fi  15235  cnmptc  15525  dvexp  15954  dvexp2  15955  dvmptidcn  15957  dvmptccn  15958  dvmptid  15959  dvmptc  15960  dvmptfsum  15968  dvef  15970  elply2  15978  plyconst  15988  plycolemc  16001  nninfall  17275  nninfsellemeqinf  17282  nninfnfiinf  17289  exmidsbthrlem  17290