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Theorem cnmptc 22265
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 5591 . 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 21886 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑃𝑌) → (𝑋 × {𝑃}) ∈ (𝐽 Cn 𝐾))
62, 3, 4, 5syl3anc 1368 . 2 (𝜑 → (𝑋 × {𝑃}) ∈ (𝐽 Cn 𝐾))
71, 6eqeltrrid 2919 1 (𝜑 → (𝑥𝑋𝑃) ∈ (𝐽 Cn 𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2114  {csn 4539  cmpt 5122   × cxp 5530  cfv 6334  (class class class)co 7140  TopOnctopon 21513   Cn ccn 21827
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 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-fv 6342  df-ov 7143  df-oprab 7144  df-mpo 7145  df-1st 7675  df-2nd 7676  df-map 8395  df-topgen 16708  df-top 21497  df-topon 21514  df-cn 21830  df-cnp 21831
This theorem is referenced by:  cnmpt2c  22273  xkoinjcn  22290  txconn  22292  imasnopn  22293  imasncld  22294  imasncls  22295  istgp2  22694  tmdmulg  22695  tmdgsum  22698  tmdlactcn  22705  clsnsg  22713  tgpt0  22722  tlmtgp  22799  nmcn  23447  fsumcn  23473  expcn  23475  divccn  23476  cncfmptc  23515  cdivcncf  23524  iirevcn  23533  iihalf1cn  23535  iihalf2cn  23537  icchmeo  23544  evth  23562  evth2  23563  pcocn  23620  pcopt  23625  pcopt2  23626  pcoass  23627  csscld  23851  clsocv  23852  dvcnvlem  24577  plycn  24856  psercn2  25016  resqrtcn  25336  sqrtcn  25337  atansopn  25516  efrlim  25553  ipasslem7  28617  occllem  29084  rmulccn  31245  cxpcncf1  31940  txsconnlem  32561  cvxpconn  32563  cvmlift2lem2  32625  cvmlift2lem3  32626  cvmliftphtlem  32638  sinccvglem  32989  knoppcnlem10  33915  areacirclem2  35104  fprodcn  42181
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