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Theorem cnmptid 22269
 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 5889 . 2 ( I ↾ 𝑋) = (𝑥𝑋𝑥)
2 cnmptid.j . . 3 (𝜑𝐽 ∈ (TopOn‘𝑋))
3 idcn 21865 . . 3 (𝐽 ∈ (TopOn‘𝑋) → ( I ↾ 𝑋) ∈ (𝐽 Cn 𝐽))
42, 3syl 17 . 2 (𝜑 → ( I ↾ 𝑋) ∈ (𝐽 Cn 𝐽))
51, 4eqeltrrid 2898 1 (𝜑 → (𝑥𝑋𝑥) ∈ (𝐽 Cn 𝐽))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2112   ↦ cmpt 5113   I cid 5427   ↾ cres 5525  ‘cfv 6328  (class class class)co 7139  TopOnctopon 21518   Cn ccn 21832 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 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445 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 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-map 8395  df-top 21502  df-topon 21519  df-cn 21835 This theorem is referenced by:  xkoinjcn  22295  txconn  22297  imasnopn  22298  imasncld  22299  imasncls  22300  pt1hmeo  22414  istgp2  22699  tmdmulg  22700  tmdlactcn  22710  clsnsg  22718  tgpt0  22727  tlmtgp  22804  nmcn  23452  expcn  23480  divccn  23481  cncfmptid  23521  cdivcncf  23529  iirevcn  23538  iihalf1cn  23540  iihalf2cn  23542  icchmeo  23549  evth2  23568  pcocn  23625  pcopt  23630  pcopt2  23631  pcoass  23632  csscld  23856  clsocv  23857  dvcnvlem  24582  resqrtcn  25341  sqrtcn  25342  efrlim  25558  ipasslem7  28622  occllem  29089  hmopidmchi  29937  rmulccn  31279  cxpcncf1  31974  cvxpconn  32597  cvmlift2lem2  32659  cvmlift2lem3  32660  cvmliftphtlem  32672  knoppcnlem10  33949  cxpcncf2  42528
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