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Theorem cnmptid 21873
 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 2065 . . . . . 6 (𝑥 = 𝑦𝑦 = 𝑥)
21opabbii 4953 . . . . 5 {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦} = {⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝑥}
3 df-id 5261 . . . . 5 I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
4 mptv 4986 . . . . 5 (𝑥 ∈ V ↦ 𝑥) = {⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝑥}
52, 3, 43eqtr4i 2812 . . . 4 I = (𝑥 ∈ V ↦ 𝑥)
65reseq1i 5638 . . 3 ( I ↾ 𝑋) = ((𝑥 ∈ V ↦ 𝑥) ↾ 𝑋)
7 ssv 3844 . . . 4 𝑋 ⊆ V
8 resmpt 5699 . . . 4 (𝑋 ⊆ V → ((𝑥 ∈ V ↦ 𝑥) ↾ 𝑋) = (𝑥𝑋𝑥))
97, 8ax-mp 5 . . 3 ((𝑥 ∈ V ↦ 𝑥) ↾ 𝑋) = (𝑥𝑋𝑥)
106, 9eqtri 2802 . 2 ( I ↾ 𝑋) = (𝑥𝑋𝑥)
11 cnmptid.j . . 3 (𝜑𝐽 ∈ (TopOn‘𝑋))
12 idcn 21469 . . 3 (𝐽 ∈ (TopOn‘𝑋) → ( I ↾ 𝑋) ∈ (𝐽 Cn 𝐽))
1311, 12syl 17 . 2 (𝜑 → ( I ↾ 𝑋) ∈ (𝐽 Cn 𝐽))
1410, 13syl5eqelr 2864 1 (𝜑 → (𝑥𝑋𝑥) ∈ (𝐽 Cn 𝐽))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1601   ∈ wcel 2107  Vcvv 3398   ⊆ wss 3792  {copab 4948   ↦ cmpt 4965   I cid 5260   ↾ cres 5357  ‘cfv 6135  (class class class)co 6922  TopOnctopon 21122   Cn ccn 21436 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-map 8142  df-top 21106  df-topon 21123  df-cn 21439 This theorem is referenced by:  xkoinjcn  21899  txconn  21901  imasnopn  21902  imasncld  21903  imasncls  21904  pt1hmeo  22018  istgp2  22303  tmdmulg  22304  tmdlactcn  22314  clsnsg  22321  tgpt0  22330  tlmtgp  22407  nmcn  23055  expcn  23083  divccn  23084  cncfmptid  23123  cdivcncf  23128  iirevcn  23137  iihalf1cn  23139  iihalf2cn  23141  icchmeo  23148  evth2  23167  pcocn  23224  pcopt  23229  pcopt2  23230  pcoass  23231  csscld  23455  clsocv  23456  dvcnvlem  24176  resqrtcn  24930  sqrtcn  24931  efrlim  25148  ipasslem7  28263  occllem  28734  hmopidmchi  29582  rmulccn  30572  cxpcncf1  31275  cvxpconn  31823  cvmlift2lem2  31885  cvmlift2lem3  31886  cvmliftphtlem  31898  knoppcnlem10  33075  cxpcncf2  41041
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