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Theorem cnmptid 22197
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 5911 . 2 ( I ↾ 𝑋) = (𝑥𝑋𝑥)
2 cnmptid.j . . 3 (𝜑𝐽 ∈ (TopOn‘𝑋))
3 idcn 21793 . . 3 (𝐽 ∈ (TopOn‘𝑋) → ( I ↾ 𝑋) ∈ (𝐽 Cn 𝐽))
42, 3syl 17 . 2 (𝜑 → ( I ↾ 𝑋) ∈ (𝐽 Cn 𝐽))
51, 4eqeltrrid 2915 1 (𝜑 → (𝑥𝑋𝑥) ∈ (𝐽 Cn 𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2105  cmpt 5137   I cid 5452  cres 5550  cfv 6348  (class class class)co 7145  TopOnctopon 21446   Cn ccn 21760
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-map 8397  df-top 21430  df-topon 21447  df-cn 21763
This theorem is referenced by:  xkoinjcn  22223  txconn  22225  imasnopn  22226  imasncld  22227  imasncls  22228  pt1hmeo  22342  istgp2  22627  tmdmulg  22628  tmdlactcn  22638  clsnsg  22645  tgpt0  22654  tlmtgp  22731  nmcn  23379  expcn  23407  divccn  23408  cncfmptid  23447  cdivcncf  23452  iirevcn  23461  iihalf1cn  23463  iihalf2cn  23465  icchmeo  23472  evth2  23491  pcocn  23548  pcopt  23553  pcopt2  23554  pcoass  23555  csscld  23779  clsocv  23780  dvcnvlem  24500  resqrtcn  25257  sqrtcn  25258  efrlim  25474  ipasslem7  28540  occllem  29007  hmopidmchi  29855  rmulccn  31070  cxpcncf1  31765  cvxpconn  32386  cvmlift2lem2  32448  cvmlift2lem3  32449  cvmliftphtlem  32461  knoppcnlem10  33738  cxpcncf2  42059
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