Theorem fconst 6567

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 5616	. . 3 ⊢ (𝐴 × {𝐵}) = (𝑥 ∈ 𝐴 ↦ 𝐵)
3	1, 2	fnmpti 6493	. 2 ⊢ (𝐴 × {𝐵}) Fn 𝐴
4		rnxpss 6031	. 2 ⊢ ran (𝐴 × {𝐵}) ⊆ {𝐵}
5		df-f 6361	. 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 2114  Vcvv 3496   ⊆ wss 3938  {csn 4569   × cxp 5555  ran crn 5558   Fn wfn 6352  ⟶wf 6353

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rab 3149  df-v 3498  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-fun 6359  df-fn 6360  df-f 6361

This theorem is referenced by:  fconstg  6568  fodomr  8670  ofsubeq0  11637  ser0f  13426  hashgval  13696  hashinf  13698  hashfxnn0  13700  prodf1f  15250  pwssplit1  19833  psrbag0  20276  xkofvcn  22294  rrx0el  24003  ibl0  24389  dvcmul  24543  dvcmulf  24544  dvexp  24552  elqaalem3  24912  basellem7  25666  basellem9  25668  axlowdimlem8  26737  axlowdimlem9  26738  axlowdimlem10  26739  axlowdimlem11  26740  axlowdimlem12  26741  0oo  28568  occllem  29082  ho01i  29607  nlelchi  29840  hmopidmchi  29930  eulerpartlemt  31631  plymul02  31818  breprexpnat  31907  noetalem3  33221  fullfunfnv  33409  fullfunfv  33410  poimirlem16  34910  poimirlem19  34913  poimirlem23  34917  poimirlem24  34918  poimirlem25  34919  poimirlem28  34922  poimirlem29  34923  poimirlem30  34924  poimirlem31  34925  poimirlem32  34926  ftc1anclem5  34973  lfl0f  36207  diophrw  39363  pwssplit4  39696  ofsubid  40663  dvsconst  40669  dvsid  40670  binomcxplemnn0  40688  binomcxplemnotnn0  40695  aacllem  44909