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

Theorem cnmptid 21274
 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 1932 . . . . . 6 (𝑥 = 𝑦𝑦 = 𝑥)
21opabbii 4649 . . . . 5 {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦} = {⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝑥}
3 dfid3 4954 . . . . 5 I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
4 mptv 4679 . . . . 5 (𝑥 ∈ V ↦ 𝑥) = {⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝑥}
52, 3, 43eqtr4i 2642 . . . 4 I = (𝑥 ∈ V ↦ 𝑥)
65reseq1i 5313 . . 3 ( I ↾ 𝑋) = ((𝑥 ∈ V ↦ 𝑥) ↾ 𝑋)
7 ssv 3588 . . . 4 𝑋 ⊆ V
8 resmpt 5369 . . . 4 (𝑋 ⊆ V → ((𝑥 ∈ V ↦ 𝑥) ↾ 𝑋) = (𝑥𝑋𝑥))
97, 8ax-mp 5 . . 3 ((𝑥 ∈ V ↦ 𝑥) ↾ 𝑋) = (𝑥𝑋𝑥)
106, 9eqtri 2632 . 2 ( I ↾ 𝑋) = (𝑥𝑋𝑥)
11 cnmptid.j . . 3 (𝜑𝐽 ∈ (TopOn‘𝑋))
12 idcn 20871 . . 3 (𝐽 ∈ (TopOn‘𝑋) → ( I ↾ 𝑋) ∈ (𝐽 Cn 𝐽))
1311, 12syl 17 . 2 (𝜑 → ( I ↾ 𝑋) ∈ (𝐽 Cn 𝐽))
1410, 13syl5eqelr 2693 1 (𝜑 → (𝑥𝑋𝑥) ∈ (𝐽 Cn 𝐽))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ⊆ wss 3540  {copab 4642   ↦ cmpt 4643   I cid 4948   ↾ cres 5040  ‘cfv 5804  (class class class)co 6549  TopOnctopon 20518   Cn ccn 20838 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-map 7746  df-top 20521  df-topon 20523  df-cn 20841 This theorem is referenced by:  xkoinjcn  21300  txcon  21302  imasnopn  21303  imasncld  21304  imasncls  21305  pt1hmeo  21419  istgp2  21705  tmdmulg  21706  tmdlactcn  21716  clsnsg  21723  tgpt0  21732  tlmtgp  21809  nmcn  22455  expcn  22483  divccn  22484  cncfmptid  22523  cdivcncf  22528  iirevcn  22537  iihalf1cn  22539  iihalf2cn  22541  icchmeo  22548  evth2  22567  pcocn  22625  pcopt  22630  pcopt2  22631  pcoass  22632  csscld  22856  clsocv  22857  dvcnvlem  23543  resqrtcn  24290  sqrtcn  24291  efrlim  24496  ipasslem7  27075  occllem  27546  hmopidmchi  28394  rmulccn  29302  cvxpcon  30478  cvmlift2lem2  30540  cvmlift2lem3  30541  cvmliftphtlem  30553  knoppcnlem10  31662  cxpcncf2  38786
 Copyright terms: Public domain W3C validator