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Theorem fconst 5988
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 5074 . . 3 (𝐴 × {𝐵}) = (𝑥𝐴𝐵)
31, 2fnmpti 5920 . 2 (𝐴 × {𝐵}) Fn 𝐴
4 rnxpss 5470 . 2 ran (𝐴 × {𝐵}) ⊆ {𝐵}
5 df-f 5793 . 2 ((𝐴 × {𝐵}):𝐴⟶{𝐵} ↔ ((𝐴 × {𝐵}) Fn 𝐴 ∧ ran (𝐴 × {𝐵}) ⊆ {𝐵}))
63, 4, 5mpbir2an 956 1 (𝐴 × {𝐵}):𝐴⟶{𝐵}
Colors of variables: wff setvar class
Syntax hints:  wcel 1976  Vcvv 3172  wss 3539  {csn 4124   × cxp 5025  ran crn 5028   Fn wfn 5784  wf 5785
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4711  ax-pr 4827
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4942  df-xp 5033  df-rel 5034  df-cnv 5035  df-co 5036  df-dm 5037  df-rn 5038  df-fun 5791  df-fn 5792  df-f 5793
This theorem is referenced by:  fconstg  5989  fodomr  7973  ofsubeq0  10866  ser0f  12673  hashgval  12939  hashinf  12941  hashf  12943  prodf1f  14411  pwssplit1  18828  psrbag0  19263  xkofvcn  21244  ibl0  23303  dvcmul  23457  dvcmulf  23458  dvexp  23466  elqaalem3  23824  basellem7  24557  basellem9  24559  axlowdimlem8  25574  axlowdimlem9  25575  axlowdimlem10  25576  axlowdimlem11  25577  axlowdimlem12  25578  0oo  26821  occllem  27339  ho01i  27864  nlelchi  28097  hmopidmchi  28187  eulerpartlemt  29553  plymul02  29742  fullfunfnv  31016  fullfunfv  31017  poimirlem16  32378  poimirlem19  32381  poimirlem23  32385  poimirlem24  32386  poimirlem25  32387  poimirlem28  32390  poimirlem29  32391  poimirlem30  32392  poimirlem31  32393  poimirlem32  32394  ftc1anclem5  32442  lfl0f  33157  diophrw  36123  pwssplit4  36460  ofsubid  37328  dvsconst  37334  dvsid  37335  binomcxplemnn0  37353  binomcxplemnotnn0  37360  aacllem  42298
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