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Theorem cnmptid 21177
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
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 equcom 1895 . . . . . 6 (𝑥 = 𝑦𝑦 = 𝑥)
21opabbii 4547 . . . . 5 {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦} = {⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝑥}
3 dfid3 4848 . . . . 5 I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
4 mptv 4577 . . . . 5 (𝑥 ∈ V ↦ 𝑥) = {⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝑥}
52, 3, 43eqtr4i 2546 . . . 4 I = (𝑥 ∈ V ↦ 𝑥)
65reseq1i 5204 . . 3 ( I ↾ 𝑋) = ((𝑥 ∈ V ↦ 𝑥) ↾ 𝑋)
7 ssv 3492 . . . 4 𝑋 ⊆ V
8 resmpt 5260 . . . 4 (𝑋 ⊆ V → ((𝑥 ∈ V ↦ 𝑥) ↾ 𝑋) = (𝑥𝑋𝑥))
97, 8ax-mp 5 . . 3 ((𝑥 ∈ V ↦ 𝑥) ↾ 𝑋) = (𝑥𝑋𝑥)
106, 9eqtri 2536 . 2 ( I ↾ 𝑋) = (𝑥𝑋𝑥)
11 cnmptid.j . . 3 (𝜑𝐽 ∈ (TopOn‘𝑋))
12 idcn 20774 . . 3 (𝐽 ∈ (TopOn‘𝑋) → ( I ↾ 𝑋) ∈ (𝐽 Cn 𝐽))
1311, 12syl 17 . 2 (𝜑 → ( I ↾ 𝑋) ∈ (𝐽 Cn 𝐽))
1410, 13syl5eqelr 2597 1 (𝜑 → (𝑥𝑋𝑥) ∈ (𝐽 Cn 𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1474  wcel 1938  Vcvv 3077  wss 3444  {copab 4540  cmpt 4541   I cid 4842  cres 4934  cfv 5689  (class class class)co 6426  TopOnctopon 20421   Cn ccn 20741
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-8 1940  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-sep 4607  ax-nul 4616  ax-pow 4668  ax-pr 4732  ax-un 6723
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ne 2686  df-ral 2805  df-rex 2806  df-rab 2809  df-v 3079  df-sbc 3307  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-nul 3778  df-if 3940  df-pw 4013  df-sn 4029  df-pr 4031  df-op 4035  df-uni 4271  df-br 4482  df-opab 4542  df-mpt 4543  df-id 4847  df-xp 4938  df-rel 4939  df-cnv 4940  df-co 4941  df-dm 4942  df-rn 4943  df-res 4944  df-ima 4945  df-iota 5653  df-fun 5691  df-fn 5692  df-f 5693  df-f1 5694  df-fo 5695  df-f1o 5696  df-fv 5697  df-ov 6429  df-oprab 6430  df-mpt2 6431  df-map 7622  df-top 20424  df-topon 20426  df-cn 20744
This theorem is referenced by:  xkoinjcn  21203  txcon  21205  imasnopn  21206  imasncld  21207  imasncls  21208  pt1hmeo  21322  istgp2  21608  tmdmulg  21609  tmdlactcn  21619  clsnsg  21626  tgpt0  21635  tlmtgp  21712  nmcn  22364  expcn  22406  divccn  22407  cncfmptid  22446  cdivcncf  22451  iirevcn  22460  iihalf1cn  22462  iihalf2cn  22464  icchmeo  22471  evth2  22490  pcocn  22548  pcopt  22553  pcopt2  22554  pcoass  22555  csscld  22720  clsocv  22721  dvcnvlem  23418  resqrtcn  24177  sqrtcn  24178  efrlim  24383  ipasslem7  26853  occllem  27334  hmopidmchi  28182  rmulccn  29098  cvxpcon  30323  cvmlift2lem2  30385  cvmlift2lem3  30386  cvmliftphtlem  30398  knoppcnlem10  31496  cxpcncf2  38686
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