Theorem fconst 6560

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 5609	. . 3 ⊢ (𝐴 × {𝐵}) = (𝑥 ∈ 𝐴 ↦ 𝐵)
3	1, 2	fnmpti 6486	. 2 ⊢ (𝐴 × {𝐵}) Fn 𝐴
4		rnxpss 6024	. 2 ⊢ ran (𝐴 × {𝐵}) ⊆ {𝐵}
5		df-f 6354	. 2 ⊢ ((𝐴 × {𝐵}):𝐴⟶{𝐵} ↔ ((𝐴 × {𝐵}) Fn 𝐴 ∧ ran (𝐴 × {𝐵}) ⊆ {𝐵}))
6	3, 4, 5	mpbir2an 709	1 ⊢ (𝐴 × {𝐵}):𝐴⟶{𝐵}

Syntax hints:   ∈ wcel 2110  Vcvv 3495   ⊆ wss 3936  {csn 4561   × cxp 5548  ran crn 5551   Fn wfn 6345  ⟶wf 6346

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-sep 5196  ax-nul 5203  ax-pr 5322

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rab 3147  df-v 3497  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4562  df-pr 4564  df-op 4568  df-br 5060  df-opab 5122  df-mpt 5140  df-id 5455  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-fun 6352  df-fn 6353  df-f 6354

This theorem is referenced by:  fconstg  6561  fodomr  8662  ofsubeq0  11629  ser0f  13417  hashgval  13687  hashinf  13689  hashfxnn0  13691  prodf1f  15242  pwssplit1  19825  psrbag0  20268  xkofvcn  22286  rrx0el  23995  ibl0  24381  dvcmul  24535  dvcmulf  24536  dvexp  24544  elqaalem3  24904  basellem7  25658  basellem9  25660  axlowdimlem8  26729  axlowdimlem9  26730  axlowdimlem10  26731  axlowdimlem11  26732  axlowdimlem12  26733  0oo  28560  occllem  29074  ho01i  29599  nlelchi  29832  hmopidmchi  29922  eulerpartlemt  31624  plymul02  31811  breprexpnat  31900  noetalem3  33214  fullfunfnv  33402  fullfunfv  33403  poimirlem16  34902  poimirlem19  34905  poimirlem23  34909  poimirlem24  34910  poimirlem25  34911  poimirlem28  34914  poimirlem29  34915  poimirlem30  34916  poimirlem31  34917  poimirlem32  34918  ftc1anclem5  34965  lfl0f  36199  diophrw  39349  pwssplit4  39682  ofsubid  40649  dvsconst  40655  dvsid  40656  binomcxplemnn0  40674  binomcxplemnotnn0  40681  aacllem  44895