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Theorem cnmptc 21513
Description: A constant function is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
Hypotheses
Ref Expression
cnmptid.j (𝜑𝐽 ∈ (TopOn‘𝑋))
cnmptc.k (𝜑𝐾 ∈ (TopOn‘𝑌))
cnmptc.p (𝜑𝑃𝑌)
Assertion
Ref Expression
cnmptc (𝜑 → (𝑥𝑋𝑃) ∈ (𝐽 Cn 𝐾))
Distinct variable groups:   𝜑,𝑥   𝑥,𝐽   𝑥,𝑋   𝑥,𝑌   𝑥,𝐾   𝑥,𝑃

Proof of Theorem cnmptc
StepHypRef Expression
1 fconstmpt 5197 . 2 (𝑋 × {𝑃}) = (𝑥𝑋𝑃)
2 cnmptid.j . . 3 (𝜑𝐽 ∈ (TopOn‘𝑋))
3 cnmptc.k . . 3 (𝜑𝐾 ∈ (TopOn‘𝑌))
4 cnmptc.p . . 3 (𝜑𝑃𝑌)
5 cnconst2 21135 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑌) → (𝑋 × {𝑃}) ∈ (𝐽 Cn 𝐾))
62, 3, 4, 5syl3anc 1366 . 2 (𝜑 → (𝑋 × {𝑃}) ∈ (𝐽 Cn 𝐾))
71, 6syl5eqelr 2735 1 (𝜑 → (𝑥𝑋𝑃) ∈ (𝐽 Cn 𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2030  {csn 4210  cmpt 4762   × cxp 5141  cfv 5926  (class class class)co 6690  TopOnctopon 20763   Cn ccn 21076
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-map 7901  df-topgen 16151  df-top 20747  df-topon 20764  df-cn 21079  df-cnp 21080
This theorem is referenced by:  cnmpt2c  21521  xkoinjcn  21538  txconn  21540  imasnopn  21541  imasncld  21542  imasncls  21543  istgp2  21942  tmdmulg  21943  tmdgsum  21946  tmdlactcn  21953  clsnsg  21960  tgpt0  21969  tlmtgp  22046  nmcn  22694  fsumcn  22720  expcn  22722  divccn  22723  cncfmptc  22761  cdivcncf  22767  iirevcn  22776  iihalf1cn  22778  iihalf2cn  22780  icchmeo  22787  evth  22805  evth2  22806  pcocn  22863  pcopt  22868  pcopt2  22869  pcoass  22870  csscld  23094  clsocv  23095  dvcnvlem  23784  plycn  24062  psercn2  24222  resqrtcn  24535  sqrtcn  24536  atansopn  24704  efrlim  24741  ipasslem7  27819  occllem  28290  rmulccn  30102  cxpcncf1  30801  txsconnlem  31348  cvxpconn  31350  cvmlift2lem2  31412  cvmlift2lem3  31413  cvmliftphtlem  31425  sinccvglem  31692  knoppcnlem10  32617  areacirclem2  33631  fprodcn  40150
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