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Theorem fconst 5592
Description: A cross product with a singleton is a constant function. (Contributed by NM, 14-Aug-1999.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
Hypothesis
Ref Expression
fconst.1  |-  B  e. 
_V
Assertion
Ref Expression
fconst  |-  ( A  X.  { B }
) : A --> { B }

Proof of Theorem fconst
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 fconst.1 . . 3  |-  B  e. 
_V
2 fconstmpt 4884 . . 3  |-  ( A  X.  { B }
)  =  ( x  e.  A  |->  B )
31, 2fnmpti 5536 . 2  |-  ( A  X.  { B }
)  Fn  A
4 rnxpss 5264 . 2  |-  ran  ( A  X.  { B }
)  C_  { B }
5 df-f 5421 . 2  |-  ( ( A  X.  { B } ) : A --> { B }  <->  ( ( A  X.  { B }
)  Fn  A  /\  ran  ( A  X.  { B } )  C_  { B } ) )
63, 4, 5mpbir2an 887 1  |-  ( A  X.  { B }
) : A --> { B }
Colors of variables: wff set class
Syntax hints:    e. wcel 1721   _Vcvv 2920    C_ wss 3284   {csn 3778    X. cxp 4839   ran crn 4842    Fn wfn 5412   -->wf 5413
This theorem is referenced by:  fconstg  5593  fodomr  7221  ofsubeq0  9957  ser0f  11335  hashgval  11580  hashinf  11582  hashf  11584  psrbag0  16513  xkofvcn  17673  ibl0  19635  dvcmul  19787  dvcmulf  19788  dvexp  19796  elqaalem3  20195  basellem7  20826  basellem9  20828  0oo  22247  occllem  22762  ho01i  23288  nlelchi  23521  hmopidmchi  23611  prodf1f  25177  fullfunfnv  25703  fullfunfv  25704  axlowdimlem8  25796  axlowdimlem9  25797  axlowdimlem10  25798  axlowdimlem11  25799  axlowdimlem12  25800  diophrw  26711  pwssplit1  27060  pwssplit4  27063  ofsubid  27413  dvsconst  27419  dvsid  27420  lfl0f  29556
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pr 4367
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-fun 5419  df-fn 5420  df-f 5421
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