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Theorem cnmptid 12523
 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 1683 . . . . . 6 (𝑥 = 𝑦𝑦 = 𝑥)
21opabbii 4004 . . . . 5 {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦} = {⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝑥}
3 df-id 4225 . . . . 5 I = {⟨𝑥, 𝑦⟩ ∣ 𝑥 = 𝑦}
4 mptv 4034 . . . . 5 (𝑥 ∈ V ↦ 𝑥) = {⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝑥}
52, 3, 43eqtr4i 2171 . . . 4 I = (𝑥 ∈ V ↦ 𝑥)
65reseq1i 4826 . . 3 ( I ↾ 𝑋) = ((𝑥 ∈ V ↦ 𝑥) ↾ 𝑋)
7 ssv 3125 . . . 4 𝑋 ⊆ V
8 resmpt 4878 . . . 4 (𝑋 ⊆ V → ((𝑥 ∈ V ↦ 𝑥) ↾ 𝑋) = (𝑥𝑋𝑥))
97, 8ax-mp 5 . . 3 ((𝑥 ∈ V ↦ 𝑥) ↾ 𝑋) = (𝑥𝑋𝑥)
106, 9eqtri 2161 . 2 ( I ↾ 𝑋) = (𝑥𝑋𝑥)
11 cnmptid.j . . 3 (𝜑𝐽 ∈ (TopOn‘𝑋))
12 idcn 12454 . . 3 (𝐽 ∈ (TopOn‘𝑋) → ( I ↾ 𝑋) ∈ (𝐽 Cn 𝐽))
1311, 12syl 14 . 2 (𝜑 → ( I ↾ 𝑋) ∈ (𝐽 Cn 𝐽))
1410, 13eqeltrrid 2228 1 (𝜑 → (𝑥𝑋𝑥) ∈ (𝐽 Cn 𝐽))
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1332   ∈ wcel 1481  Vcvv 2690   ⊆ wss 3077  {copab 3997   ↦ cmpt 3998   I cid 4220   ↾ cres 4552  ‘cfv 5134  (class class class)co 5785  TopOnctopon 12250   Cn ccn 12427 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4055  ax-pow 4107  ax-pr 4141  ax-un 4365  ax-setind 4462 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2692  df-sbc 2915  df-csb 3009  df-dif 3079  df-un 3081  df-in 3083  df-ss 3090  df-pw 3518  df-sn 3539  df-pr 3540  df-op 3542  df-uni 3746  df-iun 3824  df-br 3939  df-opab 3999  df-mpt 4000  df-id 4225  df-xp 4556  df-rel 4557  df-cnv 4558  df-co 4559  df-dm 4560  df-rn 4561  df-res 4562  df-ima 4563  df-iota 5099  df-fun 5136  df-fn 5137  df-f 5138  df-f1 5139  df-fo 5140  df-f1o 5141  df-fv 5142  df-ov 5788  df-oprab 5789  df-mpo 5790  df-1st 6049  df-2nd 6050  df-map 6555  df-top 12238  df-topon 12251  df-cn 12430 This theorem is referenced by:  imasnopn  12541
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