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

Theorem cnrmnrm 23283
Description: A completely normal space is normal. (Contributed by Mario Carneiro, 26-Aug-2015.)
Assertion
Ref Expression
cnrmnrm (𝐽 ∈ CNrm → 𝐽 ∈ Nrm)

Proof of Theorem cnrmnrm
StepHypRef Expression
1 eqid 2727 . . 3 𝐽 = 𝐽
21restid 17420 . 2 (𝐽 ∈ CNrm → (𝐽t 𝐽) = 𝐽)
3 uniexg 7749 . . 3 (𝐽 ∈ CNrm → 𝐽 ∈ V)
4 cnrmi 23282 . . 3 ((𝐽 ∈ CNrm ∧ 𝐽 ∈ V) → (𝐽t 𝐽) ∈ Nrm)
53, 4mpdan 685 . 2 (𝐽 ∈ CNrm → (𝐽t 𝐽) ∈ Nrm)
62, 5eqeltrrd 2829 1 (𝐽 ∈ CNrm → 𝐽 ∈ Nrm)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2098  Vcvv 3471   cuni 4910  (class class class)co 7424  t crest 17407  Nrmcnrm 23232  CNrmccnrm 23233
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698  ax-rep 5287  ax-sep 5301  ax-nul 5308  ax-pow 5367  ax-pr 5431  ax-un 7744
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-reu 3373  df-rab 3429  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-iun 5000  df-br 5151  df-opab 5213  df-mpt 5234  df-id 5578  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-f1 6556  df-fo 6557  df-f1o 6558  df-fv 6559  df-ov 7427  df-oprab 7428  df-mpo 7429  df-rest 17409  df-cnrm 23240
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator