Theorem fconst 6656

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 5648	. . 3 ⊢ (𝐴 × {𝐵}) = (𝑥 ∈ 𝐴 ↦ 𝐵)
3	1, 2	fnmpti 6572	. 2 ⊢ (𝐴 × {𝐵}) Fn 𝐴
4		rnxpss 6072	. 2 ⊢ ran (𝐴 × {𝐵}) ⊆ {𝐵}
5		df-f 6434	. 2 ⊢ ((𝐴 × {𝐵}):𝐴⟶{𝐵} ↔ ((𝐴 × {𝐵}) Fn 𝐴 ∧ ran (𝐴 × {𝐵}) ⊆ {𝐵}))
6	3, 4, 5	mpbir2an 707	1 ⊢ (𝐴 × {𝐵}):𝐴⟶{𝐵}

Syntax hints:   ∈ wcel 2109  Vcvv 3430   ⊆ wss 3891  {csn 4566   × cxp 5586  ran crn 5589   Fn wfn 6425  ⟶wf 6426

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-sep 5226  ax-nul 5233  ax-pr 5355

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-ral 3070  df-rab 3074  df-v 3432  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-sn 4567  df-pr 4569  df-op 4573  df-br 5079  df-opab 5141  df-mpt 5162  df-id 5488  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-rn 5599  df-fun 6432  df-fn 6433  df-f 6434

This theorem is referenced by:  fconstg  6657  fodomr  8880  ofsubeq0  11953  ser0f  13757  hashgval  14028  hashinf  14030  hashfxnn0  14032  prodf1f  15585  pwssplit1  20302  psrbag0  21251  xkofvcn  22816  rrx0el  24543  ibl0  24932  dvcmul  25089  dvcmulf  25090  dvexp  25098  elqaalem3  25462  basellem7  26217  basellem9  26219  axlowdimlem8  27298  axlowdimlem9  27299  axlowdimlem10  27300  axlowdimlem11  27301  axlowdimlem12  27302  0oo  29130  occllem  29644  ho01i  30169  nlelchi  30402  hmopidmchi  30492  eulerpartlemt  32317  plymul02  32504  breprexpnat  32593  noetasuplem4  33918  fullfunfnv  34227  fullfunfv  34228  poimirlem16  35772  poimirlem19  35775  poimirlem23  35779  poimirlem24  35780  poimirlem25  35781  poimirlem28  35784  poimirlem29  35785  poimirlem30  35786  poimirlem31  35787  poimirlem32  35788  ftc1anclem5  35833  lfl0f  37062  diophrw  40561  pwssplit4  40894  ofsubid  41895  dvsconst  41901  dvsid  41902  binomcxplemnn0  41920  binomcxplemnotnn0  41927  aacllem  46457