Theorem fconst 6550

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 5583	. . 3 ⊢ (𝐴 × {𝐵}) = (𝑥 ∈ 𝐴 ↦ 𝐵)
3	1, 2	fnmpti 6474	. 2 ⊢ (𝐴 × {𝐵}) Fn 𝐴
4		rnxpss 6001	. 2 ⊢ ran (𝐴 × {𝐵}) ⊆ {𝐵}
5		df-f 6339	. 2 ⊢ ((𝐴 × {𝐵}):𝐴⟶{𝐵} ↔ ((𝐴 × {𝐵}) Fn 𝐴 ∧ ran (𝐴 × {𝐵}) ⊆ {𝐵}))
6	3, 4, 5	mpbir2an 710	1 ⊢ (𝐴 × {𝐵}):𝐴⟶{𝐵}

Syntax hints:   ∈ wcel 2111  Vcvv 3409   ⊆ wss 3858  {csn 4522   × cxp 5522  ran crn 5525   Fn wfn 6330  ⟶wf 6331

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pr 5298

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-v 3411  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-sn 4523  df-pr 4525  df-op 4529  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-fun 6337  df-fn 6338  df-f 6339

This theorem is referenced by:  fconstg  6551  fodomr  8690  ofsubeq0  11671  ser0f  13473  hashgval  13743  hashinf  13745  hashfxnn0  13747  prodf1f  15296  pwssplit1  19899  psrbag0  20823  xkofvcn  22384  rrx0el  24098  ibl0  24486  dvcmul  24643  dvcmulf  24644  dvexp  24652  elqaalem3  25016  basellem7  25771  basellem9  25773  axlowdimlem8  26842  axlowdimlem9  26843  axlowdimlem10  26844  axlowdimlem11  26845  axlowdimlem12  26846  0oo  28671  occllem  29185  ho01i  29710  nlelchi  29943  hmopidmchi  30033  eulerpartlemt  31857  plymul02  32044  breprexpnat  32133  noetasuplem4  33524  fullfunfnv  33819  fullfunfv  33820  poimirlem16  35375  poimirlem19  35378  poimirlem23  35382  poimirlem24  35383  poimirlem25  35384  poimirlem28  35387  poimirlem29  35388  poimirlem30  35389  poimirlem31  35390  poimirlem32  35391  ftc1anclem5  35436  lfl0f  36667  diophrw  40095  pwssplit4  40428  ofsubid  41423  dvsconst  41429  dvsid  41430  binomcxplemnn0  41448  binomcxplemnotnn0  41455  aacllem  45742