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Theorem fnconstg 6563
Description: A Cartesian product with a singleton is a constant function. (Contributed by NM, 24-Jul-2014.)
Assertion
Ref Expression
fnconstg (𝐵𝑉 → (𝐴 × {𝐵}) Fn 𝐴)

Proof of Theorem fnconstg
StepHypRef Expression
1 fconstg 6562 . 2 (𝐵𝑉 → (𝐴 × {𝐵}):𝐴⟶{𝐵})
21ffnd 6511 1 (𝐵𝑉 → (𝐴 × {𝐵}) Fn 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106  {csn 4563   × cxp 5551   Fn wfn 6346
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-ext 2796  ax-sep 5199  ax-nul 5206  ax-pr 5325
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2615  df-eu 2649  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rab 3151  df-v 3501  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-br 5063  df-opab 5125  df-mpt 5143  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-fun 6353  df-fn 6354  df-f 6355
This theorem is referenced by:  fconst2g  6964  ofc1  7425  ofc2  7426  caofid0l  7430  caofid0r  7431  caofid1  7432  caofid2  7433  fnsuppres  7851  fczsupp0  7853  fczfsuppd  8843  brwdom2  9029  cantnf0  9130  ofnegsub  11628  ofsubge0  11629  pwsplusgval  16755  pwsmulrval  16756  pwsvscafval  16759  pwsco1mhm  17981  dprdsubg  19068  pwsmgp  19290  pwssplit1  19753  frlmpwsfi  20812  frlmbas  20815  frlmvscaval  20828  islindf4  20898  tmdgsum2  22620  0plef  24188  0pledm  24189  itg1ge0  24202  mbfi1fseqlem5  24235  xrge0f  24247  itg2ge0  24251  itg2addlem  24274  bddibl  24355  dvidlem  24428  rolle  24502  dveq0  24512  dv11cn  24513  tdeglem4  24569  mdeg0  24579  fta1blem  24677  qaa  24827  basellem9  25580  ofcc  31252  ofcof  31253  eulerpartlemt  31516  noextendseq  33059  matunitlindflem1  34756  matunitlindflem2  34757  ptrecube  34760  poimirlem1  34761  poimirlem2  34762  poimirlem3  34763  poimirlem4  34764  poimirlem5  34765  poimirlem6  34766  poimirlem7  34767  poimirlem10  34770  poimirlem11  34771  poimirlem12  34772  poimirlem16  34776  poimirlem17  34777  poimirlem19  34779  poimirlem20  34780  poimirlem22  34782  poimirlem23  34783  poimirlem28  34788  poimirlem29  34789  poimirlem31  34791  poimirlem32  34792  broucube  34794  cnpwstotbnd  34944  eqlkr2  36104  pwssplit4  39551  mpaaeu  39612  rngunsnply  39635  ofdivrec  40520  dvconstbi  40528  zlmodzxzscm  44234  aacllem  44731
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