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Theorem cnmptid 21458
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 1944 . . . . . 6 (𝑥 = 𝑦𝑦 = 𝑥)
21opabbii 4715 . . . . 5 {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦} = {⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝑥}
3 dfid3 5023 . . . . 5 I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
4 mptv 4749 . . . . 5 (𝑥 ∈ V ↦ 𝑥) = {⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝑥}
52, 3, 43eqtr4i 2653 . . . 4 I = (𝑥 ∈ V ↦ 𝑥)
65reseq1i 5390 . . 3 ( I ↾ 𝑋) = ((𝑥 ∈ V ↦ 𝑥) ↾ 𝑋)
7 ssv 3623 . . . 4 𝑋 ⊆ V
8 resmpt 5447 . . . 4 (𝑋 ⊆ V → ((𝑥 ∈ V ↦ 𝑥) ↾ 𝑋) = (𝑥𝑋𝑥))
97, 8ax-mp 5 . . 3 ((𝑥 ∈ V ↦ 𝑥) ↾ 𝑋) = (𝑥𝑋𝑥)
106, 9eqtri 2643 . 2 ( I ↾ 𝑋) = (𝑥𝑋𝑥)
11 cnmptid.j . . 3 (𝜑𝐽 ∈ (TopOn‘𝑋))
12 idcn 21055 . . 3 (𝐽 ∈ (TopOn‘𝑋) → ( I ↾ 𝑋) ∈ (𝐽 Cn 𝐽))
1311, 12syl 17 . 2 (𝜑 → ( I ↾ 𝑋) ∈ (𝐽 Cn 𝐽))
1410, 13syl5eqelr 2705 1 (𝜑 → (𝑥𝑋𝑥) ∈ (𝐽 Cn 𝐽))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1482  wcel 1989  Vcvv 3198  wss 3572  {copab 4710  cmpt 4727   I cid 5021  cres 5114  cfv 5886  (class class class)co 6647  TopOnctopon 20709   Cn ccn 21022
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-sbc 3434  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-ov 6650  df-oprab 6651  df-mpt2 6652  df-map 7856  df-top 20693  df-topon 20710  df-cn 21025
This theorem is referenced by:  xkoinjcn  21484  txconn  21486  imasnopn  21487  imasncld  21488  imasncls  21489  pt1hmeo  21603  istgp2  21889  tmdmulg  21890  tmdlactcn  21900  clsnsg  21907  tgpt0  21916  tlmtgp  21993  nmcn  22641  expcn  22669  divccn  22670  cncfmptid  22709  cdivcncf  22714  iirevcn  22723  iihalf1cn  22725  iihalf2cn  22727  icchmeo  22734  evth2  22753  pcocn  22811  pcopt  22816  pcopt2  22817  pcoass  22818  csscld  23042  clsocv  23043  dvcnvlem  23733  resqrtcn  24484  sqrtcn  24485  efrlim  24690  ipasslem7  27675  occllem  28146  hmopidmchi  28994  rmulccn  29959  cxpcncf1  30658  cvxpconn  31209  cvmlift2lem2  31271  cvmlift2lem3  31272  cvmliftphtlem  31284  knoppcnlem10  32476  cxpcncf2  39882
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