Theorem fconst 6306

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 5368	. . 3 ⊢ (𝐴 × {𝐵}) = (𝑥 ∈ 𝐴 ↦ 𝐵)
3	1, 2	fnmpti 6233	. 2 ⊢ (𝐴 × {𝐵}) Fn 𝐴
4		rnxpss 5783	. 2 ⊢ ran (𝐴 × {𝐵}) ⊆ {𝐵}
5		df-f 6105	. 2 ⊢ ((𝐴 × {𝐵}):𝐴⟶{𝐵} ↔ ((𝐴 × {𝐵}) Fn 𝐴 ∧ ran (𝐴 × {𝐵}) ⊆ {𝐵}))
6	3, 4, 5	mpbir2an 703	1 ⊢ (𝐴 × {𝐵}):𝐴⟶{𝐵}

Syntax hints:   ∈ wcel 2157  Vcvv 3385   ⊆ wss 3769  {csn 4368   × cxp 5310  ran crn 5313   Fn wfn 6096  ⟶wf 6097

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-fun 6103  df-fn 6104  df-f 6105

This theorem is referenced by:  fconstg  6307  fodomr  8353  ofsubeq0  11309  ser0f  13108  hashgval  13373  hashinf  13375  hashfxnn0  13377  prodf1f  14961  pwssplit1  19380  psrbag0  19816  xkofvcn  21816  ibl0  23894  dvcmul  24048  dvcmulf  24049  dvexp  24057  elqaalem3  24417  basellem7  25165  basellem9  25167  axlowdimlem8  26186  axlowdimlem9  26187  axlowdimlem10  26188  axlowdimlem11  26189  axlowdimlem12  26190  0oo  28169  occllem  28687  ho01i  29212  nlelchi  29445  hmopidmchi  29535  eulerpartlemt  30949  plymul02  31141  breprexpnat  31232  noetalem3  32378  fullfunfnv  32566  fullfunfv  32567  poimirlem16  33914  poimirlem19  33917  poimirlem23  33921  poimirlem24  33922  poimirlem25  33923  poimirlem28  33926  poimirlem29  33927  poimirlem30  33928  poimirlem31  33929  poimirlem32  33930  ftc1anclem5  33977  lfl0f  35090  diophrw  38108  pwssplit4  38444  ofsubid  39305  dvsconst  39311  dvsid  39312  binomcxplemnn0  39330  binomcxplemnotnn0  39337  aacllem  43349