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Theorem fconst 6129
Description: A Cartesian 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 ∈ V
Assertion
Ref Expression
fconst (𝐴 × {𝐵}):𝐴⟶{𝐵}

Proof of Theorem fconst
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 fconst.1 . . 3 𝐵 ∈ V
2 fconstmpt 5197 . . 3 (𝐴 × {𝐵}) = (𝑥𝐴𝐵)
31, 2fnmpti 6060 . 2 (𝐴 × {𝐵}) Fn 𝐴
4 rnxpss 5601 . 2 ran (𝐴 × {𝐵}) ⊆ {𝐵}
5 df-f 5930 . 2 ((𝐴 × {𝐵}):𝐴⟶{𝐵} ↔ ((𝐴 × {𝐵}) Fn 𝐴 ∧ ran (𝐴 × {𝐵}) ⊆ {𝐵}))
63, 4, 5mpbir2an 975 1 (𝐴 × {𝐵}):𝐴⟶{𝐵}
Colors of variables: wff setvar class
Syntax hints:  wcel 2030  Vcvv 3231  wss 3607  {csn 4210   × cxp 5141  ran crn 5144   Fn wfn 5921  wf 5922
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-fun 5928  df-fn 5929  df-f 5930
This theorem is referenced by:  fconstg  6130  fodomr  8152  ofsubeq0  11055  ser0f  12894  hashgval  13160  hashinf  13162  hashfxnn0  13164  hashfOLD  13166  prodf1f  14668  pwssplit1  19107  psrbag0  19542  xkofvcn  21535  ibl0  23598  dvcmul  23752  dvcmulf  23753  dvexp  23761  elqaalem3  24121  basellem7  24858  basellem9  24860  axlowdimlem8  25874  axlowdimlem9  25875  axlowdimlem10  25876  axlowdimlem11  25877  axlowdimlem12  25878  0oo  27772  occllem  28290  ho01i  28815  nlelchi  29048  hmopidmchi  29138  eulerpartlemt  30561  plymul02  30751  breprexpnat  30840  noetalem3  31990  fullfunfnv  32178  fullfunfv  32179  poimirlem16  33555  poimirlem19  33558  poimirlem23  33562  poimirlem24  33563  poimirlem25  33564  poimirlem28  33567  poimirlem29  33568  poimirlem30  33569  poimirlem31  33570  poimirlem32  33571  ftc1anclem5  33619  lfl0f  34674  diophrw  37639  pwssplit4  37976  ofsubid  38840  dvsconst  38846  dvsid  38847  binomcxplemnn0  38865  binomcxplemnotnn0  38872  aacllem  42875
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