Theorem fconst 6232

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.

Step	Hyp	Ref	Expression
1		fconst.1	. . 3 ⊢ 𝐵 ∈ V
2		fconstmpt 5302	. . 3 ⊢ (𝐴 × {𝐵}) = (𝑥 ∈ 𝐴 ↦ 𝐵)
3	1, 2	fnmpti 6161	. 2 ⊢ (𝐴 × {𝐵}) Fn 𝐴
4		rnxpss 5706	. 2 ⊢ ran (𝐴 × {𝐵}) ⊆ {𝐵}
5		df-f 6034	. 2 ⊢ ((𝐴 × {𝐵}):𝐴⟶{𝐵} ↔ ((𝐴 × {𝐵}) Fn 𝐴 ∧ ran (𝐴 × {𝐵}) ⊆ {𝐵}))
6	3, 4, 5	mpbir2an 690	1 ⊢ (𝐴 × {𝐵}):𝐴⟶{𝐵}

Syntax hints:   ∈ wcel 2145  Vcvv 3351   ⊆ wss 3723  {csn 4317   × cxp 5248  ran crn 5251   Fn wfn 6025  ⟶wf 6026

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-fun 6032  df-fn 6033  df-f 6034

This theorem is referenced by:  fconstg  6233  fodomr  8271  ofsubeq0  11223  ser0f  13061  hashgval  13324  hashinf  13326  hashfxnn0  13328  hashfOLD  13330  prodf1f  14831  pwssplit1  19272  psrbag0  19709  xkofvcn  21708  ibl0  23773  dvcmul  23927  dvcmulf  23928  dvexp  23936  elqaalem3  24296  basellem7  25034  basellem9  25036  axlowdimlem8  26050  axlowdimlem9  26051  axlowdimlem10  26052  axlowdimlem11  26053  axlowdimlem12  26054  0oo  27984  occllem  28502  ho01i  29027  nlelchi  29260  hmopidmchi  29350  eulerpartlemt  30773  plymul02  30963  breprexpnat  31052  noetalem3  32202  fullfunfnv  32390  fullfunfv  32391  poimirlem16  33757  poimirlem19  33760  poimirlem23  33764  poimirlem24  33765  poimirlem25  33766  poimirlem28  33769  poimirlem29  33770  poimirlem30  33771  poimirlem31  33772  poimirlem32  33773  ftc1anclem5  33820  lfl0f  34876  diophrw  37846  pwssplit4  38183  ofsubid  39047  dvsconst  39053  dvsid  39054  binomcxplemnn0  39072  binomcxplemnotnn0  39079  aacllem  43073