MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fconst6 Structured version   Visualization version   GIF version

Theorem fconst6 6781
Description: A constant function as a mapping. (Contributed by Jeff Madsen, 30-Nov-2009.) (Revised by Mario Carneiro, 22-Apr-2015.)
Hypothesis
Ref Expression
fconst6.1 𝐵𝐶
Assertion
Ref Expression
fconst6 (𝐴 × {𝐵}):𝐴𝐶

Proof of Theorem fconst6
StepHypRef Expression
1 fconst6.1 . 2 𝐵𝐶
2 fconst6g 6780 . 2 (𝐵𝐶 → (𝐴 × {𝐵}):𝐴𝐶)
31, 2ax-mp 5 1 (𝐴 × {𝐵}):𝐴𝐶
Colors of variables: wff setvar class
Syntax hints:  wcel 2105  {csn 4628   × cxp 5674  wf 6539
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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-fun 6545  df-fn 6546  df-f 6547
This theorem is referenced by:  ramz  16963  psrlidm  21743  psrbag0  21843  00ply1bas  21983  ply1plusgfvi  21985  mbfpos  25401  i1f0  25437  noxp1o  27403  axlowdimlem1  28468  axlowdimlem7  28474  axlowdim1  28485  hlim0  30756  0cnfn  31501  0lnfn  31506  circlemethnat  33952  circlevma  33953  poimirlem29  36821  poimirlem30  36822  poimirlem31  36823  poimir  36825  broucube  36826  expgrowth  43397
  Copyright terms: Public domain W3C validator