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Theorem fconst 5658
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 4950 . . 3  |-  ( A  X.  { B }
)  =  ( x  e.  A  |->  B )
31, 2fnmpti 5602 . 2  |-  ( A  X.  { B }
)  Fn  A
4 rnxpss 5330 . 2  |-  ran  ( A  X.  { B }
)  C_  { B }
5 df-f 5487 . 2  |-  ( ( A  X.  { B } ) : A --> { B }  <->  ( ( A  X.  { B }
)  Fn  A  /\  ran  ( A  X.  { B } )  C_  { B } ) )
63, 4, 5mpbir2an 888 1  |-  ( A  X.  { B }
) : A --> { B }
Colors of variables: wff set class
Syntax hints:    e. wcel 1727   _Vcvv 2962    C_ wss 3306   {csn 3838    X. cxp 4905   ran crn 4908    Fn wfn 5478   -->wf 5479
This theorem is referenced by:  fconstg  5659  fodomr  7287  ofsubeq0  10028  ser0f  11407  hashgval  11652  hashinf  11654  hashf  11656  psrbag0  16585  xkofvcn  17747  ibl0  19707  dvcmul  19861  dvcmulf  19862  dvexp  19870  elqaalem3  20269  basellem7  20900  basellem9  20902  0oo  22321  occllem  22836  ho01i  23362  nlelchi  23595  hmopidmchi  23685  prodf1f  25251  fullfunfnv  25822  fullfunfv  25823  axlowdimlem8  25919  axlowdimlem9  25920  axlowdimlem10  25921  axlowdimlem11  25922  axlowdimlem12  25923  ftc1anclem5  26322  diophrw  26855  pwssplit1  27203  pwssplit4  27206  ofsubid  27556  dvsconst  27562  dvsid  27563  lfl0f  29965
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pr 4432
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-rab 2720  df-v 2964  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-sn 3844  df-pr 3845  df-op 3847  df-br 4238  df-opab 4292  df-mpt 4293  df-id 4527  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-fun 5485  df-fn 5486  df-f 5487
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