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

Theorem cnmptid 23669
Description: The identity function is continuous. (Contributed by Mario Carneiro, 5-May-2014.) (Revised by Mario Carneiro, 22-Aug-2015.)
Hypothesis
Ref Expression
cnmptid.j (𝜑𝐽 ∈ (TopOn‘𝑋))
Assertion
Ref Expression
cnmptid (𝜑 → (𝑥𝑋𝑥) ∈ (𝐽 Cn 𝐽))
Distinct variable groups:   𝜑,𝑥   𝑥,𝐽   𝑥,𝑋

Proof of Theorem cnmptid
StepHypRef Expression
1 mptresid 6069 . 2 ( I ↾ 𝑋) = (𝑥𝑋𝑥)
2 cnmptid.j . . 3 (𝜑𝐽 ∈ (TopOn‘𝑋))
3 idcn 23265 . . 3 (𝐽 ∈ (TopOn‘𝑋) → ( I ↾ 𝑋) ∈ (𝐽 Cn 𝐽))
42, 3syl 17 . 2 (𝜑 → ( I ↾ 𝑋) ∈ (𝐽 Cn 𝐽))
51, 4eqeltrrid 2846 1 (𝜑 → (𝑥𝑋𝑥) ∈ (𝐽 Cn 𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2108  cmpt 5225   I cid 5577  cres 5687  cfv 6561  (class class class)co 7431  TopOnctopon 22916   Cn ccn 23232
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8868  df-top 22900  df-topon 22917  df-cn 23235
This theorem is referenced by:  xkoinjcn  23695  txconn  23697  imasnopn  23698  imasncld  23699  imasncls  23700  pt1hmeo  23814  istgp2  24099  tmdmulg  24100  tmdlactcn  24110  clsnsg  24118  tgpt0  24127  tlmtgp  24204  nmcn  24866  expcn  24896  divccn  24897  expcnOLD  24898  divccnOLD  24899  cncfmptid  24939  cdivcncf  24947  iirevcn  24957  iihalf1cn  24959  iihalf1cnOLD  24960  iihalf2cn  24962  iihalf2cnOLD  24963  icchmeo  24971  icchmeoOLD  24972  evth2  24992  pcocn  25050  pcopt  25055  pcopt2  25056  pcoass  25057  csscld  25283  clsocv  25284  dvcnvlem  26014  resqrtcn  26792  sqrtcn  26793  efrlim  27012  efrlimOLD  27013  ipasslem7  30855  occllem  31322  hmopidmchi  32170  rmulccn  33927  cxpcncf1  34610  cvxpconn  35247  cvmlift2lem2  35309  cvmlift2lem3  35310  cvmliftphtlem  35322  knoppcnlem10  36503  cxpcncf2  45914
  Copyright terms: Public domain W3C validator